arxiv:hep-ph/9606301v1 11 jun 1996 · 2018-09-21 · arxiv:hep-ph/9606301v1 11 jun 1996 ect*-96-010...

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ECT*-96-010TK 96 18

NORDITA 96/35 N,P

Workshop on

THE STANDARD MODEL

AT LOW ENERGIES

ECT*, Trento, Italy, April 29 — May 10, 1996

ABSTRACT BOOKLET

These are short proceedings of the workshop on “The Standard Model at Low Energies” heldat ECT* in Trento, Italy, from April 29 to May 10, 1996. The workshop concentrated on ChiralPerturbation Theory in its various settings. Included are a one page abstract with referencesper speaker and a listing of some review papers of relevance to the field.

J. BijnensNORDITA, Blegdamsvej 17,

DK-2100 Copenhagen Ø, Denmark

U.-G. MeißnerUniversitat Bonn, Institut fur Theoretische Kernphysik, Nussallee 14-16,

D-53115 Bonn, Germany

http://arxiv.org/abs/hep-ph/9606301v1

1 Introduction

The field of Chiral Perturbation Theory is a very active one. It was therefore felt that anothertopical workshop was needed. This meeting followed the series of workshops in Ringberg (Ger-many), 1988, Dobog/’ok/”o (Hungary), 1991, and Karrebæksminde (Denmark), 1993, whichwere all three of the size of about 50 participants. We were lucky enough to obtain fundingfrom the European Center for Theoretical Studies in Nuclear Physics and related Areas, theECT*, for this workshop.

The meeting itself took place in Trento in the spring of 1996. There were a lot of discussionsand talks. The present abstract booklet is meant as a guide into the literature and providesabstracts and main references of the presentations given. Since most results will be publishedelsewhere, a full traditional type of proceedings seemed unnecessary.

We would like to take the opportunity to thank the ECT*, its director, Ben Mottelson, andits board of directors for general support and enthusiasm for this meeting. We would also liketo thank the secretaries Cristina Costa at ECT* and Anne Lumholdt at NORDITA for takingcare of the administrative work involved. Finally we thank the participants for making this avery pleasant and lively meeting.

Unfortunately, we cannot thank the weatherman since it rained most of the workshop, butthis way the presence of most participants at the talks was guaranteed.

There exists by now a rather large series of main references related to Chiral PerturbationTheory. We give below a short and subjective list of the most recent ones and those consideredclassics. Then the list of participants and their electronic mail addresses follows. The programof the meeting and the individual abstracts with their references round off this abstract booklet.The latter are given in the same order as the talks were given. Speakers are indicated in boldface.

Johan Bijnens and Ulf-G. Meißner

1

2 A short guide to review literature

Chiral Perturbation Theory grew out of current algebra, and it soon was realized that certainterms beyond the lowest order were also uniquely defined. This early work and references toearlier review papers can be found in [1]. Weinberg then proposed a systematic method in [2],later systematized and extended to use the external field method in the classic papers by Gasserand Leutwyler [3,4], which, according to Howard Georgi, everybody should put under his/herpillow before he/she goes to sleep. The field has since then extended a lot and relatively recentreview papers are: Ref.[5] with an emphasis on the anomalous sector, Ref.[6] giving a generaloverview over the vast field of applications in various areas of physics, Ref.[7] on mesons andbaryons, and Ref.[8] on baryons and multibaryon processes. In addition there are books byGeorgi[9], which, however, does not cover the standard approach, including the terms in thelagrangian at higher order and a more recent one by Donoghue, Golowich and Holstein[10].

There are also the lectures available on the archives by E. de Rafael [11] and A. Pich[12]as well as numerous others. The references to the previous meetings are [13,14,15]. There arealso the MIT meeting [16] and the DAΦNE handbook [17] as useful references.

References[1] H. Pagels, Phys. Rep. 16 (1975) 219[2] S. Weinberg, Physica 96A (1979) 327[3] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142[4] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465[5] J. Bijnens, Int. J. Mod. Phys. A8 (1993) 3045[6] U.-G. Meißner, Rep. Prog. Phys. 56 (1993) 903[7] G. Ecker, Prog. Nucl. Part. Phys. 35 (1995) 1[8] V. Bernard, N. Kaiser and U.-G. Meißner, Int. J. Mod. Phys. E4 (1995) 193[9] H. Georgi, Weak Interactions and Modern Particle Theory, 1984, Benjamin/Cummings.[10] J. Donoghue, E. Golowich and B. Holstein, Dynamics of the Standard Model, CambridgeUniversity Press.[11] E. de Rafael, hep-ph/9502254, lectures at the TASI-94 Summer School, ed. J.F. Donoghue,World Scientific, Singapore, 1995.[12] A. Pich, Rep. Prog. Phys. 58 (1995) 563, hep-ph 9505231[13] A. Buras, J.-M. Gerard and W. Huber (eds.), Nucl. Phys. B (Proc. Suppl) 7A (1989)[14] U.-G. Meißner (ed.), Effective Field Theories of the Standard Model, World Scientific,Singapore, 1992[15] J. Bijnens, NORDITA-93/73.[16] A. Bernstein and B. Holstein (eds.), Chiral Dynamics : Theory and Experiment, SpringerVerlag, 1995.[17] L. Maiani, G. Pancheri and N. Paver (eds.), The Second DAΦNE Physics Handbook, 1995,SIS-Pubblicazione dei Laboratori Nazionali di Frascati, P.O.Box 13, I-00044 Frascati, Italy.

2

http://arxiv.org/abs/hep-ph/9502254


3 Participants and their email

S.Beane [email protected]

V.Bernard [email protected]

J.Bijnens [email protected]

B.Borasoy [email protected]

M.Butler [email protected]

P.Buettiker [email protected]

G.Colangelo [email protected]

F.Cornet [email protected]

G.D’Ambrosio [email protected]

D.Drechsel [email protected]

S.Duerr [email protected]

G.Ecker [email protected]

H.Forkel [email protected]

A.Gall [email protected]

J.Gasser [email protected]

E.Golowich [email protected]

C.Hanhart [email protected]

T.Hannah [email protected]

T.Hemmert [email protected]

G.Isidori [email protected]

N.Kaiser [email protected]

J.Kambor [email protected]

M.Knecht [email protected]

S.Krewald [email protected]

M.Lutz [email protected]

J.Matias [email protected]

F.Meier [email protected]

U.Meissner [email protected]

B.Moussallam [email protected]

G.Mueller [email protected]

E.Pallante [email protected]

M.Pennington [email protected]

A.Perez [email protected]

T.Pham [email protected]

J.Portoles [email protected]

J.Prades [email protected]

E.de Rafael [email protected]

M.Sainio [email protected]

J.Schechter [email protected]

S.Scherer [email protected]

N.Scoccola [email protected]

A.Smilga [email protected]

R.Springer [email protected]

J.Steele [email protected]

J.Stern [email protected]

3

D.Toublan [email protected]

R.Urech [email protected]

G.Valencia [email protected]

U.Van Kolck [email protected]

T.Waas [email protected]

T.Watabe [email protected]

H.Weigel [email protected]

C.Wiesendanger [email protected]

A.Wirzba [email protected]

4

4 The program as it finally ended

week 1Monday 29/4

morning empty for arrival purposes14.30 J. Bijnens Administrative and Other Arrangements15.00 U. Meißner Introductory remarks about CHPTTuesday 30/49.15 Paul Buttiker The Chiral coupling constants l1 and l2 from ππ10.15 Joseph Schechter Simple description of ππ scattering to 1 GeV11.00 Coffee Break11.30 Bachir Moussallam Sum rules in ππ scattering12.00 Lunch15.00 Veronique Bernard πN → ππN in CHPT15.45 Stefan Scherer Extension of the CHPT Meson Lagrangian to Order p6

16.30 Coffee Break17.00 Eugene Golowich Two-loop analysis of Vector-Current and Axialvector-current

Propagator, a progress report.17.45Wednesday 1/510.00 Ubirajara van Kolck Isospin as an Accidental Symmetry10.45 Coffee Break11.15 Christoph Hanhart π-threshold production in pp-collisions11.45 Lunch15.00 Frank Meier Quantum Corrections to Baryon Properties in Chiral

Soliton Models15.30 Andreas Wirzba S-wave pion propagation in dense isosymmetric nuclear matter16.15 Coffee Break16.45 Jan Stern Experimental signature of quark condensates17.30Thursday 2/59.15 Jurg Gasser Pion polarizabilities to two loops10.15 Coffee Break11.00 Tri Nang Pham Chiral Lagrangian with vector and axial vector mesons for

π+-π0 mass difference11.30 Antonio Perez Electromagnetic mass differences of pions and kaons12.00 Lunch15.00 Matthias Lutz Chiral symmetry and many-nucleon systems15.45 Guido Muller Renormalization of the Pion-Nucleon Lagrangian to order p4

16.15 Coffee Break16.45 Norbert Kaiser Neutral pion photo– and electroproduction17.30Friday 3/59.15 Dieter Dreschel Pion Photoproduction of the Nucleon -results from

Dispersion Theory10.00 Bugra Borasoy Baryon Masses to Second Order in the Quark Masses

5

10.45 Coffee Break11.15 James V. Steele Master Approach in the Nucleon Sector12.00 Lunch15.00 A. Smilga Scalar susceptibility in QCD and in multiflavor Schwinger model15.45 Jan Stern Quark condensate and density of states16.15week 2Monday 6/59.45 Gilberto Colangelo Elastic ππ scattering to Two Loops10.30 Coffee Break11.00 Marc Knecht The ππ scattering amplitude to two loops11.45 Dominique Toublan Low Energy Sum Rules For Pion-Pion Scattering

and Threshold Parameters12.15 Lunch15.00 Herbert Weigel Heavy Quark Solitons15.45 Coffee Break16.15 Eduardo de Rafael Low Energy QCD in the large Nc limit17.00Tuesday 7/59.15 Michael Pennington Dispersive analysis of χPT predictions10.00 Christian Final State Interactions and Khuri-Treiman Equations

Wiesendanger in η → 3π10.45 Coffee Break11.15 Thomas Hemmert ∆(1232) in Chiral Perturbation Theory12.00 Lunch15.00 Joachim Kambor Resonance Saturation in the Baryon Sector15.45 Coffee Break16.15 Silas Beane Novel algebraic consequences of chiral symmetry17.00Wednesday 8/59.15 Elisabetta Pallante Hadronic Contributions to the muon g-2: an updated analysis10.00 Joaquim Prades Some Hadronic Matrix Elements in ENJL :

BK , Dashen’s Theorem, γγ → ππ10.45 Coffee Break11.15 Res Urech On the corrections to Dashen’s theorem11.45 LunchThursday 9/59.15 Giancarlo D’Ambrosio Topics in Radiative Non-leptonic Kaon Decays10.00 Gino Isidori Radiative Four-Meson Amplitudes10.45 Coffee Break11.15 Gerhard Ecker Aspects of renormalization in CHPT12.00 Lunch15.00 Roxanne P. Springer Chiral Symmetry and Hypernuclei15.45 Norberto Scoccola Hyperon Electromagnetic Properties in a Soliton Model16.15 Coffee Break16.45 Teruaki Watabe Strange Contents in Nucleon; Difficulty and Approach17.30

6

Friday 10/59.15 Joaquim Matias CHPT description of the MSM: One and two loop order10.00 Stephan Duerr The covariant derivative expansion10.30 Coffee Break11.00 Thomas Waas Kaon nucleon interaction and the

Λ(1405) in dense matter11.4514.00 Johan Bijnens γγ → πππ and some comments on U(1)A.14.30 Ulf Meißner The organizers have the final word as usual.

7

Remarks on CHPT and EFTs

Ulf–G. MeißnerUniversitat Bonn, Institut fur Theoretische Kernphysik

D-53115 Bonn, Germany

In this introductory talk, I discuss certain aspects of chiral perturbation theory (CHPT),which is the effective field theory (EFT) of the standard model. In EFTs, based on the powercounting first introduced by Weinberg [1], one considers tree and loop graphs of the light dofs(the heavy ones being integrated out) and the effective Lagrangian is organized according tothe chiral dimension (or number of derivatives). Often it is argued that it makes no sense toconsider loops in such an approach since the loop momenta can not be considered small as itis the case for the external momenta. Clearly, one could invent a scheme in which one wouldcut all the pion momenta to be less than the typical scale of the heavy degrees of freedom.Alternatively, using dimensional regularization, one chooses the associated scale to be of theorder of the heavy mass scale. This effectively suppresses the high momentum components inthe loops. However, this high–energy information is not lost, it is encoded in the values of theassociated low energy constants (LECs) appearing to the order one is working.

A second remark concerns the use of dispersion relations to not only extend the range ofapplicability of the chiral predictions but also to sharpen these at the very low energy end.Having calculated e.g. the imaginary part of the scalar pion form factor and the elastic ππscattering amplitude to one loop [2] allows one to write a dispersive representation to two loopaccuracy for the scalar form factor Γπ with a number of subtractions to guarantee convergence[3]. These subtraction constants play a role similar to the LECs in the corresponding “real”two loop calculation. While the analytic structure of the two approaches is identical, the latterone contains more information. First, the LECs can be related to other processes and second,only in certain circumstances the enhancement of certain LECs due to IR logs can be unraveledin the dispersive approach, e.g. the finiteness of the Γπ in the chiral limit reveals the lnM2

π

dependence of d2. Terms of the type M2π lnM

2π can not be found by such means. If one only

considers a certain observable or process, like Γπ or ππ → ππ, a next to next to leading ordercalculation is certainly much easier done in the dispersive approach. Also, the equivalent of aone loop calculation can be done without ever performing a loop integral. The prize one paysis the loss of information described above.

References[1] S. Weinberg, Physica 96A (1979) 327[2] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142[3] J. Gasser and Ulf–G. Meißner, Nucl. Phys. B357 (1991) 90.

8

Chiral Coupling Constants from ππ Phase Shifts

B. Ananthanarayan and P. ButtikerInstitut fur Theoretische Physik, Universitat Bern, CH–3012 Bern, Switzerland

ChPT [1] provides the low energy effective theory of the standard model describing interactionsinvolving hadronic degrees of freedom. It is a nonrenormalizable theory; additional couplingconstants have to be introduced at each order of the derivative or momentum expansion. Atleading order O(p2) there are two such constants, the pion decay constant Fπ and the pion massmπ. At next to leading order O(p4) there are ten more constants. Four of them l1, l2, l3 andl4 enter the ππ scattering amplitude. As a result, the threshold parameters can be expressedin terms of these as well. In the past the coupling constants l1 and l2 have been fixed fromexperimental values for the D-wave scattering lengths [2] or from an analysis of Kl4 decays [3].On the other hand ππ scattering has been studied in great detail in axiomatic field theory [4].Fixed-t dispersion relations have been established in the axiomatic framework and properties ofcrossing and analyticity have been exploited to establish the Roy equations, a system of integralequations for the partial wave amplitudes [5,6]. Here we report on a direct determination of thecoupling constants from the existing phase shift data [7,8] by performing a Roy equation fit toit when a00 is restricted to the range predicted by ChPT. Using certain properties of the chiralamplitude [9], we write down a dispersion representation with a certain number of subtractionsconsistent with O(p4) accuracy, where the subtraction constants are expressed in terms of thechiral coupling constants. The fixed-t dispersion relations of axiomatic field theory are alsorewritten in a form whereby a direct comparison with the chiral dispersive representation canbe made while the subtraction constants are now computed in terms of physical partial waves,produced by the Roy equation fit. As an example we cite l1 = −1.7 ± 0.15 and l2 ≈ 5.0 forthe one-loop coupling constants. Our method is powerful enough to be extended in a straightforward manner to determine two-loop coupling constants.

References[1] J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984)[2] M. M. Nagels et al., Nucl. Phys. B 147, 189 (1979)[3] C. Riggenbach, J.F. Donoghue, J. Gasser and B.R. Holstein, Phys.Rev. 4

¯3,127 (1991);

J. Bijnens, G. Colangelo and J. Gasser, Nucl. Phys. B427, 427 (1994)[4] A. Martin, “Scattering Theory: Unitarity, Analyticity and Crossing”,

Springer-Verlag, Berlin, Heidelberg, New York, 1969[5] S.M. Roy, Phys. Lett. 36B, 353 (1971)[6] J.–L. Basdevant et al., Nucl. Phys. B72, 413 (1974)[7] W. Ochs, Thesis, Ludwig-Maximilians-Universitat, Munchen, 1973[8] L. Rosselet, et al., Phys. Rev. D 15, 574 (1977)[9] J. Stern, H. Sazdjian and N. H. Fuchs, Phys. Rev. D 47, 3814 (1993)

9

Simple Description of ππ Scattering to One GeV

M. Harada, F. Sannino and J. SchechterDepartment of PhysicsSyracuse University

Syracuse, NY 13244-1130, USA

In this work, described in detail in [1], we slightly relax the extremely accurate description ofthe threshold region obtained in chiral perturbation theory in order to describe ππ scatteringall the way up to the 1 GeV region.

The present model can be viewed as an attempt to approximate the leading “Born” term ofthe 1/Nc expansion of the ππ amplitude in QCD. It is known that such an amplitude containscontact terms and an infinite number of resonance exchanges. We truncate the resonances tothose in the energy region up to about 1.4 GeV. We get the amplitude from a chiral Lagrangianso that crossing symmetry is automatically satisfied. Since the leading 1/Nc amplitude containssingularities (zero width resonances) and is otherwise purely real we a) restrict attention topredicting the real part of the amplitude b)regularize the amplitude at the pole positions insuch a way that “local unitarity”(near the resonance poles) is maintained.

It is found that the resulting amplitude satisfies the unitarity bounds in addition to crossingsymmetry for the I = l = 0 channel (the difficult one) up to 1.2 GeV. The components are 1)the “current algebra” contact term, 2) the ρ exchange diagrams, 3) a broad scalar resonanceat about 560 MeV, and 4) the f0(980) with its associated Ramsauer-Townsend effect. Thecontributions of the “next group” of resonances (comprising the f2(1275),the f0(1300) and theρ(1450) tend to cancel each other and thus do not disturb the nice picture in this energy range.A similar mechanism is observed in the off diagonal process ππ → KK.

References[1] M. Harada, F. Sannino and J. Schechter, hep-ph/9511335. See also F. Sannino and J.Schechter, Phys Rev D52,96(1995).

10


Determination of Two-loop ππ Scattering Amplitude Parameters

Bachir MoussallamDivision de Physique Theorique1, Institut de Physique Nucleaire

F-91406 Orsay Cedex, France

The expansion of the ππ amplitude to two-loop chiral order has recently been worked out[1][2], it is expressed in terms of elementary functions of the Mandelstam variables and of sixparameters. These involve combinations of O(p4) as well as O(p6) low energy constants andchiral logarithms [2]. We have shown that four of these parameters obey sum rules which allowan accurate determination based on existing ππ scattering data at medium energies (

√s be-

tween 0.5 to 2 GeV)[3]. These sum rules are obtained by matching chiral perturbation theorywith dispersion relations, a technique which was used in a variety of applications in recentyears. A specific feature of the elastic ππ amplitude, is the invariance under crossing (moduloa crossing matrix). This invariance gives rise to the Roy dispersive representation [4]. We haverederived and improved this representation taking into account the notion of chiral counting,dropping all contributions of chiral order higher or equal to eight. As a consequence, all thenecessary and sufficient conditions for crossing symmetry to hold can be explicitated in a sim-ple way (which was not the case in the original formulation): in addition to determining thesubtraction functions up to two constants, the so called driving terms are also determined tobe polynomials and, finally, a relation is found between three integrals over high-energy data.Equating the scattering function A(s, t, u) from a) the chiral expansion and b) the Roy dis-persive representation gives the four sum rules. We have finally shown that it is by no meansnecessary to solve numerically the Roy equations in order to exploit the sum rules: in the lowenergy region, where sufficiently precise scattering data is not available, it is as efficient to usethe chiral expansion of the amplitude: taking two parameters as input ( equivalently, one coulduse a00 and a20) the four remaining ones are determined in a self consistent way by the sumrules. One of the two parameters which are left free has a particularly interesting theoreticalsignificance related to the way in which chiral symmetry is spontaneously broken in QCD: theinterested reader should consult the contribution of Marc Knecht in these proceedings.

References[1] M. Knecht, B. Moussallam, J. Stern and N. H. Fuchs, Nucl. Phys. B457 (1995) 513.[2] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M. Sainio, Phys. Lett. B374 (1996) 210.[3] M. Knecht, B. Moussallam, J. Stern and N. H. Fuchs, hep-ph 9512404, to appear in Nucl.Phys. B.[4] S.M. Roy, Phys. Lett. 36B (1971) 353, J.L. Basdevant, J.C. Le Guillou and H. Navelet,Nuovo Cimento 7A (1972) 363.

1Laboratoire de Recherche des Universites Paris XI et Paris VI associe au CNRS

11


The reaction πN → ππN in HBCHPT

Veronique Bernard, Norbert Kaiser, Ulf–G. MeißnerUniversite Louis Pasteur, Physique Theorique

F-67037 Strasbourg Cedex 2, France

In the framework of heavy baryon chiral perturbation theory (HBCHPT), we give the chiralexpansion for the πN → ππN threshold amplitudes D1 and D2 to linear and to quadratic orderin the pion mass. To linear order in the pion mass, we derive low–energy theorems (LETs) forthe two threshold amplitudes D1 and D2 which are free of unknown low–energy constants. Thenumerical predictions of these LETs work well for the reaction π+p → π+π+n but show somesignificant deviations for π−p → π0π0n as naively expected [1]. To second order in the pionmass, the theoretical results agree within one standard deviation with the empirical values [2].We notice that the effect of pion rescattering is efficiently masked by pion–nucleon rescatteringand resonance excitation, in particular due to the N⋆(1440). We find a novel N⋆ → N(ππ)Scoupling which has not been accounted for in previous phenomenological analysis. We alsoderive a relation between the two threshold amplitudes of the reaction πN → ππN and theππ S–wave scattering lengths, a00 and a20, respectively, to order O(M2

π) [2]. We show that theuncertainties mostly related to resonance excitation make an accurate determination of the ππscattering length a00 from the ππN threshold amplitudes at present very difficult. From theexisting data, we deduce a00 = 0.21± 0.07 where the error does not include (presumably large)contributions at O(M3

π). The situation is different in the ππ isospin two final state. Here, thechiral series converges and one finds a20 = −0.031± 0.007 somewhat smaller than the two–loopchiral perturbation theory prediction. These results could be used to determine l3 which is theparameter directly related to the size of the condensate. However, at the present state of theart l3 will be given with a rather large error bar. We also point out that previous analysis ofthe same data using the Olsson–Turner model can not be trusted [3].

References[1] V. Bernard, N. Kaiser and U. Meißner, Phys. Lett. B332 (1994) 415; (E) B338 (1994) 520.[2] V. Bernard, N. Kaiser and U. Meißner, Nucl. Phys. B457 (1995) 147.[3] M.G. Olsson, U. Meißner, N. Kaiser and V. Bernard, πN Newsletter 10 (1995) 201

12

Extension of the chiral perturbation theory meson Lagrangian toorder p6

Harold W. Fearing (1) and Stefan Scherer (2)(1) TRIUMF, Vancouver, Canada

(2) Institut fur Kernphysik, Mainz, Germany

We discuss the most general chirally invariant Lagrangian L6 for the meson sector at order p6

within the framework of standard SU(3) chiral perturbation theory [1]. The result [2] providesan extension of the well-known Gasser-Leutwyler Lagrangian L4 to one higher order, includingas well all the odd-intrinsic-parity terms in the Lagrangian. We have developed a systematicstrategy so as to get all the independent terms and eliminate the redundant ones in an efficientway. For that purpose we have introduced a twofold hierarchy in terms of a) the number ofcovariant derivatives and b) the number of traces contained in an expression. This procedureallows to eliminate terms in favor of ones lower in the hierarchy without actually working outthe explicit and often extremely complicated relations connecting the corresponding terms. Weexplain how field transformations can be used to identify redundant terms which are propor-tional to the lowest-order equation of motion [3]. The claim to have obtained the most generalLagrangian relies on this systematic construction and on the elimination of the redundantquantities using using relations of which we are aware, rather than on a general formal proofof either completeness or independence.

The end result involves more than hundred terms which, under certain assumptions, fallinto two distinct classes of interaction terms, according to whether they are even or odd inthe number of Goldstone bosons. We have separated the final set of terms into groupings ofexpressions contributing to increasingly more complicated processes, so that one does not haveto deal with the full result when calculating p6 contributions to simple processes.

References[1] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465.[2] H. W. Fearing and S. Scherer, Phys. Rev. D53 (1996) 315.[3] S. Scherer and H. W. Fearing, Phys. Rev. D52 (1995) 6445.

13

Two-loop Analysis of Vector-current and Axialvector-currentPropagators in Chiral Perturbation Theory: A Progress Report.

Eugene GolowichUniversity of MassachusettsAmherst MA 01003 USA

In this talk, I describe a calculational program by Joachim Kambor and myself for deter-mining the low energy behaviour of propagators ∆µν

V,A,33(q2) and ∆µν

V,A,88(q2) at two-loop order

in ChPT.The vector-current part of the project was recently brought to a satisfactory conclusion

[1,2]. We determined the propagators with straightforward Feynman diagram methods bymaking appropriate use of external vector sources. At O(q4) there were three diagrams and atO(q6) there were ten. To absorb divergences and scale dependence at two-loop level requiredconstruction of appropriate counterterms from the O(q6) chiral lagrangian. Of the four suchO(q6) counterterms found, only three are independent in the vector-current sector. The finalresults, in finite, covariant, and scale-independent form, yielded a successful fit to data of thetwo-loop isospin vector spectral function for E ≤ 400 MeV. [1]

In [2], we used ‘inverse-moment’ sum rules derived in [1] to obtain phenomenological eval-uations of two of the three new O(q6) counterterms. Our analysis also yielded insights on theimportant but difficult issue regarding contributions of higher orders in the ChPT expansionto the sum rules.

Work continues [3], now on the axialvector propagators. Although the vector and axialvectorcalculations are similar in overall structure, there are several differences of detail, e.g. two-looprenormalizations of masses and decay constants appear in the axialvector systems via the mesonpole. To date, we have determined all one-particle irreducible two-loop diagrams, including theformidable ‘sunset’ diagram. We have also obtained the list of O(q6) counterterms whichcontribute to the axialvector sector.

References[1] E. Golowich and J. Kambor, Nucl. Phys. B447, (1995) 373.[2] E. Golowich and J. Kambor, Phys. Rev. D53, (1996) 2651.[3] E. Golowich and J. Kambor, work in progress.

14

Isospin as an Accidental Symmetry

U. van KolckDepartment of Physics, University of Washington

Seattle, WA 98195-1560, USA

Isospin is shown [1] to be an accidental symmetry, in the sense of being a symmetry thatis present in the effective low-energy theory (the general chiral lagrangian involving pions,nucleons, and delta isobars) in lowest order but not in the underlying theory (QC+ED).

I start by constructing the operators involving the low-energy degrees of freedom that breakchiral symmetry in the same way as quark mass and charge difference terms in QC+ED; suchoperators appear in the chiral lagrangian with coefficients proportional to powers of the up-downmass difference and the fine structure constant. I use naive dimensional power counting to showthat there are no isospin violating operators in lowest order: in most processes isospin violationwill therefore be down compared to isospin conserving terms not only by a ratio of the quarkmass difference to the sum, but also by additional powers of the ratio between the low energyof interest (Q ∼ pion mass) and the QCD scale (M ∼ rho mass). The operators are next usedto study simple processes in leading orders. It is easy to see that isospin violation in pion-pionscattering should come predominantly from explicit photon loops. Pion-nucleon scattering, onthe other hand, could in principle show “large” (i.e., not suppressed by extra powers of Q/M)isospin violation related to quark mass effects in the t-channel isoscalar amplitude, but thisis hard to measure. Turning to nuclear systems, I find that i) the leading breaking of isospincomes from the (predominantly electromagnetic) pion mass difference in the one-pion-exchangetwo-nucleon potential, an effect that still preserves charge symmetry; ii) charge symmetrybreaking is O(Q/M) smaller, and arises from an isospin-violating pion-nucleon coupling, andtwo short-range interactions, all mainly quark mass effects. (From the viewpoint of meson-exchange models, they might come respectively from π–η mixing, and ρ–ω and a1–f1 mixings[2].) Finally, the use of such a chiral lagrangian is illustrated by a computation of the pion-range, isospin-violating two-nucleon potential to third order in the chiral expansion [3]. Thisincludes, besides the tree graphs with isospin dependent pion masses and couplings mentionedabove, also one-loop diagrams with pion [2] and photon dressings at vertices and propagators,and with simultaneous pion and photon exchange.

References[1] U. van Kolck, Ph.D. Dissertation, Univ. of Texas (1993); Univ. of Washington preprintDOE/ER/40427-13-N94 (in preparation).[2] U. van Kolck, J.L. Friar, and T. Goldman, Phys. Lett. B 371 (1996) 169.[3] J.L. Friar, T. Goldman, and U. van Kolck, in preparation.

15

Threshold pion production in nucleon-nucleon collisions

C. Hanhart, J. Haidenbauer and J. SpethInstitut fur Kernphysik, Forschungszentrum Julich

Julich, Germany

We performed a momentum-space calculation of the reaction NN → NNπ near threshold,extending our earlier study [1]. The following pion production mechanisms are considered:(i) Direct emission of the pion form one of the nucleons. (ii) (s-wave) rescattering where theproduced pion first scatters off the other nucleon before its emission. (iii) Contributions frommeson-exchange currents due to the exchange of heavy mesons (σ, ω) connected to intermediateNN pairs[2]. The Bonn OBEPT potential [3] is used for the distortions in the initial and finalNN states. For the evaluation of the rescattering contribution a microscopic meson–exchangemodel of πN interaction developed recently by the Julich group [4] is utilized. A soft form factor(of monopole type) with a cut–off mass ΛπNN = 800 MeV, as suggested, e.g., by recent QCDlattice calculations [5], is employed at the pion production vertex. In the calculation of theheavy-meson-exchange contributions we us the same ω vertex parameters as in OBEPT. The σ,however, is an effective parameterization of correlated 2-πexchange (and other processes) andits strength should be different in the NN interaction and in the present case, where it couplesto NN pairs. Therefore the coupling constant of the σ is treated as free parameter. We achievea quantitative description of the measured total cross section for the reaction pp → ppπ0 nearthreshold. With the same model (and the same parameter set) we are also able to reproduce thpp → dπ+ cross section near threshold with similar quality. Note, however, that in the latterreaction the pion production via a ∆-isobar excitation could be important [6]. This process isso far neglected in our study. The reaction NN → NNπ close to threshold promises to givea deeper insight into the off–shell properties of the NN interaction as well as into the shortrange correlations of the NN force. It might also be sensitive to additional constrains fromchiral symmetry as suggested by recent investigations using chiral perturbation theory[7].

References[1] C. Hanhart, J. Haidenbauer, A. Reuber, C. Schutz and J. Speth, Phys. Lett. B358(1995)21-26.[2] T.-S. Lee and D. Riska, Phys. Rev. Lett. 70, 2237 (1993).[3] R.Machleit, K.Holinde, and Ch.Elster, Phys. Rep. 149, 1 (1987) .[4] C. Schutz, J.W. Durso, K. Holinde, and J.Speth, Phys. Rev. C 49, 2671 (1994).[5] K.F.Liu, S.J.Dong, T.Draper, and W.Wilcox, Phys. Rev. Lett. 74, 2171 .[6] J.A.Niskanen, Phys. Rev. C, 53, 526 (1996) .[7] T.Y.Park et al., Phys. Rev. C, 53, 1519 (1996) and T.D.Cohen et al., nucl-th/9512036 .

16

http://arxiv.org/abs/nucl-th/9512036

S-wave pion propagation in dense isosymmetric nuclear matter

Andreas WirzbaInstitut fur Theoretische Kernphysik, Technische Hochschule Darmstadt

Schloßgartenstr. 9, D-64289 Darmstadt, Germany

The starting point of the talk is the isoscalar S-wave interaction between pions and nucleons.The corresponding O(Q2)-heavy-baryon lagrangian of ref. [1] (in the mean-field approximationfor the nucleons) is then applied to finite baryon densities ρ. Note that nuclear correlations arethus neglected. Within the generating functional formalism of ref. [2] the effective mass of thepion in isosymmetric homogeneous nuclear matter is derived [3,4] and shown to be independentof the various off-mass-shell extension schemes (as e.g. PCAC) [3].

With the help of the corresponding generating functional the density-dependent quark con-densate 〈uu+dd〉ρ as well as the in-medium axialvector-axialvector and axialvector-pseudoscalarcorrelators are derived and the time-like pion decay constant F t

π(ρ) (which in the matter back-ground is bigger than its space-like counter part F s

π(ρ)) and the pseudoscalar coupling con-stant G∗

π(ρ) are deduced [3,4]. These quantities combine to satisfy the in-medium extensionof the Gell-Mann-Oakes-Renner relation, F t

π(ρ)2m∗

π(ρ)2 = −mq〈uu + dd〉ρ + O(Q3, ρ2), and

the density-dependent PCAC relation, F tπ(ρ)m

∗π(ρ)

2 = mqG∗π(ρ) + O(Q3, ρ2), where mq is the

SU(2)-averaged current quark mass. Furthermore the results are compatible with Migdal’s ap-proach to finite Fermi systems on the composite hadron level provided the Migdal propagatorhas been identified correctly [5].

Finally, the mean-field lagrangian sets constraints for the in-medium extension of chiralperturbation theory [4]. As the matter background selects a special Lorentz-frame, the in-medium version of ChPTh cannot satisfy Lorentz invariance, but only the left-over Euclideanrotational invariance. It should therefore be classified as a non-relativistic ChPTh of ref. [6].Indeed, the dispersion of the S-wave pion-propagation in isosymmetric nuclear matter is toorder O(Q2) the same as for the corresponding Goldstone (π) bosons of an antiferromagnet,where the spacelike “F s

π” is smaller than the time-like “F tπ” [6] as well. However, the fact

that the πN S-wave lagrangian predicts, in the mean-field approximation, the first correctionsto the isoscalar channel to be of order O(Q3) is incompatible with standard ChPTh (wherethey are of order O(Q4) whether in the relativistic or non-relativistic version). Thus the in-medium (non-relativistic) ChPTh has to be of the generalized form of ref. [7]. This is supportedby the possibility that, with increasing baryon density, the in-medium quark condensate canpotentially become so small, that the higher in-medium quark condensates cannot be neglectedany longer as it is the case in standard ChPTh in the vacuum.

References[1] V. Bernard, N. Kaiser and U.-G. Meißner, Phys. Lett. B309 (1993) 421[2] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142[3] V. Thorsson and A. Wirzba, Nucl. Phys. A 589 (1995) 633[4] A. Wirzba and V. Thorsson, Hirschegg ’95, GSI, 1995, hep-ph/9502314[5] M. Kirchbach and A. Wirzba, Nucl. Phys. A in print, nucl-th/9603017[6] H. Leutwyler, Phys. Rev D49 (1994) 3033[7] M. Knecht and J. Stern, DAPHNE physics handbook (2nd ed.), hep-ph/9411253

17


http://arxiv.org/abs/nucl-th/9603017


Experimental Signature of Quark-Antiquark Condensation in theQCD Vacuum

Jan SternDivision de Physique Theorique1, Institut de Physique Nucleaire


It is generally believed that in QCD, the spontaneous breakdown of chiral symmetry is a con-sequence of a strong quark-antiquark condensation in the vacuum. The condensate parameterB0 = −qq/F 2 is expected to be sufficiently large to insure that, for actual values of runningquark masses, the expansion of the square of the Goldstone boson masses is dominated by thefirst Gell-Mann–Oakes–Renner term. This assumption is crucial in the standard formulation[1] of CHPT, but sofar, it has not been tested experimentally. Moreover, a sound theoreticalalternative [2] in which the condensate B0 would be marginal (typically 10 times smaller thanbelieved) or even vanishing, could naturally arise in QCD and it is not excluded by any existingdata. For these reasons an experimental probe of quark condensation becomes of fundamen-tal importance. The best evidence in favour of or against a strong qq condensation wouldbe provided by new high precision low energy pipi scattering experiments [3]. ( The actualuncertainty, c.f. a00 = 0.26 ± 0.05 encompasses both alternatives of a strong and weak quarkcondensation.) Additional signature could emerge from the analysis of deviations from theGoldberger Treimann relations [4], η → 3π decays [5], γγ → π0π0 near threshold, providedcorresponding experimental data become more accurate. The question of the strength of quarkcondensation could influence on various (not yet tested ) predictions: estimates of light quarkmasses [6], estimates of ε′/ε and some issues in the B-physics, among others.

References[1] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465.[2] M. Knecht and J. Stern in The Second Daphne Physics Handbook, eds. L.Maiani,G.Pancheriand N.Paver, INFN-LNF publication, May 1995.[3] N. H. Fuchs, H. Sazdjian and J. Stern, Phys. Lett. B269 (1991) 183; Phys. Rev.D47 (1993)3814.[4] N. H. Fuchs, H. Sazdjian and J. Stern Phys. Lett. B238 (1990) 380.[5] M. Knecht and J. Novotny in preparation.[6] J.Stern, M.Knecht and N.H. Fuchs in Proceedings of the Third Workchop on the Tau-CharmFactory, Marbella (Spain), June 1993, Eds. J. Kirkby and R. Kirkby, Editions Frontieres (1993).

1Laboratoire de Recherche des Universites Paris XI et Paris VI, associe au CNRS

18

Pion polarizabilities to two loops

J. GasserInstitut fur theoretische Physik, Universitat Bern,

Sidlerstrasse 5, CH–3012 Bern

I have discussed in my talk the evaluation of the pion electric (απ) and magnetic (βπ)polarizabilities in the framework of chiral perturbation theory. The polarizabilities are obtainedby expanding the Compton amplitude near threshold in powers of the photon momenta. Theleading term in this momentum expansion is proportional to the square of the charge of thepion, whereas the coefficients of the next–to–leading–order term are determined by απ and βπ.In the chiral expansion, the polarizabilities receive their leading order contribution from termsat order p4 [1-3]. At this order, one has απ + βπ = 0 (D–wave term). Therefore, to determinethe first nonvanishing contribution to απ+ βπ, a two–loop evaluation of the Compton amplitudeis needed.

The result of this calculation in the neutral channel has been published some time ago[4], whereas the evaluation in the charged channel (which involves considerably more diagrams,because there is a tree graph contribution) has been completed recently [5]. The two low–energyconstants that enter the polarizabilities at two–loop order have been determined by resonancesaturation in these references. Chiral logarithms contribute in both channels in a nonnegligiblemanner. Burgi has also investigated the importance of various graphs in the MS scheme [5],using the sigmamodel parametrization of the U–matrix. He finds that the acnode and the boxgraphs generate substantial contributions to the polarizabilities.

References[1] J.F. Donoghue and B.R. Holstein, Phys. Rev. D40 (1989) 2378.[2] B.R. Holstein, Comm. Nucl. Part. Phys. 19 (1990) 221.[3] J. Bijnens and F. Cornet, Nucl. Phys. B296 (1988) 557.[4] S. Bellucci, J. Gasser and M.E. Sainio, Nucl. Phys. B423 (1994) 80.[5] U. Burgi, Charged pion polarizabilities to two loops, Bern University Preprint BUTP–96/01, hep–ph/9602421, to appear in Phys. Lett. B; Charged pion–pair production and pionpolarizabilities to two loops, Bern University Preprint BUTP–96/02, hep-ph/9602429.

19

http://arxiv.org/abs/hep--ph/9602421


CHIRAL LAGRANGIAN WITH VECTOR AND AXIALVECTOR MESON FOR π+ − π0 MASS DIFFERENCE

T. N. PhamCentre de Physique Theorique,

Centre National de la Recherche Scientifique, UPR A0014,

Ecole Polytechnique, 91128 Palaiseau Cedex, France

In this talk, I report on a recent work [1] on a simple derivation of the forward virtualCompton scattering off a soft pion target for use in the π+−π0 electromagnetic mass differencecalculation. References to previous works can be found in this paper.

By using a nonlinear chiral Lagrangian with vector and axial vector meson incorporated ina modified gauged chiral model, it is shown that the simple expression for the forward virtualCompton scattering on a soft pion usually obtained from Current Algebra can be derived in asimple manner. This shows also that the absence of the double pole behaviour for the Bornterms is a consequence of chiral symmetry. Though this result has also been obtained previuoslyby Ecker et al. and also more recently by Donoghue et al. in which the vector and axial vectormeson fields are treated as antisymmetric tensor representation instead of the usual four-vectorfield operator, we show that one can also derive the Current Algebra result in a simple mannerwith the conventional four-vector representation for vector and axial vector meson. We notealso that the unsubtracted dispersion relation for the ∆I = 2 amplitude can be made consistentwith the soft pion result by including also the contact term from the vector meson pole termin a modified Born term. Then it would be more convenient to use the dispersion relationapproach to calculate the π+ − π0 mass difference since terms of O(p2) can also be analysed ina straightforward manner.

References[1] T. N. Pham, Phys.Lett. 374 (1989) 205 .

20

The Electromagnetic Mass Differences of Pions and Kaons

John F. Donoghue(a) and Antonio F. Perez(a,b)

(a) Department of Physics and AstronomyUniversity of Massachusetts, Amherst, MA 01003

(b) Department of PhysicsUniversity of Cincinnati, Cinicinnati, OH

We use the Cottingham method to calculate the pion and kaon electromagnetic mass differ-ences with as few model dependent inputs as possible. The constraints of chiral symmetry atlow energy, QCD at high energy and experimental data in between are used in the dispersionrelation. We find excellent agreement with experiment for the pion mass difference. The kaonmass difference exhibits a strong violation of the lowest order predictions via Dashen’s theorem,in qualitative agreement with several other recent calculations.

References, (Partial List)[1] Aston, D. et al., Nucl. Phys. B202, pg. 21, 1982.[2] Baur, R., Urech , R., hep-ph/9508393, Feb. 22 1996.[3] Bijnens, J., Phys. Lett., B306, pg. 343, 1993.[4] Collins,J. C., Nucl. Phys. B149, pg. 90, 1979.[5] Cottingham, W. N., Annals. Phys. 25, pg. 424, 1963.[6] Das et al., Phys. Rev. Lett. 18, pg. 759, 1967.[7] Dashen, R., Phys. Rev. 183, pg. 1245, 1969.[8] Daum, C. et al., Nucl. Phys. 187, pg. 1, 1981.[9] Donoghue, J. et al., Phys. Rev., D47, pg. 2089, 1993.[10] Donoghue, J., and Golowich, E., Phys. Rev., D49, pg. 1513, 1994.[11] Ecker et al., Nucl. Phys. B321, pg. 311, 1989.[12] Ecker et al., Phys. Lett. B223, pg. 425, 1989.[13] Gasser, J. and Leutwyler, H., Annals of Phys. 158, pg. 142, 1984.[14] Gasser, J. and Leutwyler, H., Nucl. Phys. B250, pg. 465, 1985.[15] O’Donnell, P. J., Rev. Mod. Phys. 53, pg. 673, 1981.[16] Particle Data Group, Phys. Rev. D50,, 1994.[17] Socolow, R. H., Phys. Rev. 137, pg. B1221, 1965.[18] Weinberg, S., Phys. Rev. Lett. 18, pg. 507, 1967.

21


Chiral approach to NN-scattering amplitude

Matthias LutzECT∗ Villa Tambosi

Trento, Italy

Weinberg [1] and van Kolck [2] introduced a nonrelativistic perturbative scheme with consistentchiral counting rules for the nucleon-nucleon potential. In this talk we present a relativistic chiralexpansion scheme for the nucleon-nucleon scattering amplitude. It is advantageous to work withthe manifestly Lorenz covariant chiral Lagrangian where we achieve the desired nonrelativistic1/m expansion by a proper regrouping of the interaction terms in the Lagrangian. The 1/mexpansion can then be performed at the level of each individual Feynman diagram. We find thatour relativistic scheme naturally sums the non-polymomial terms in 1/m needed to reconcileproper dispersion relations and threshold behaviour.

Chiral counting rules, predicting the leading chiral power for the two-nucleon irreducibleFeynman diagrams of the Bethe-Salpeter kernel, are in full analogy to Weinberg’s countingrules for the nucleon-nucleon potential. To further derive chiral power counting rules for theNN -scattering amplitude we introduce a subtraction scheme (for a given finite cutoff Λ) at thelevel of the Bethe-Salpeter equation such that its solution, the NN -scattering amplitude, isindependent of the subtraction scheme with its characteristic scale µ. A small subtraction scaleµ, of the order of the pion mass, renders the unitary iterations of the properly subtracted pionexchanges perturbative with a well defined chiral counting rule: each intermediate two-nucleonstate generates a chiral enhancement power -1. The subtraction scale determines the relativeimportance of the unitary iterations of the 2-nucleon vertices as compared to the strengthprovided by unitary iterations of pion exchanges. We find that a small subtraction scale µcauses a strong renormalization of the local s-wave nucleon-nucleon interaction vertices at givenphysical cutoff Λ. In this case the natural s-wave bare couplings mutate into large couplings,which acquire the anomalous chiral power −1. The thus renormalized local s-wave interactionvertices pick up sufficient strength to generate naturally the deuteron bound state and thepseudo-bound state in the nucleon-nucleon scattering amplitude upon unitary iterations.

Our scattering amplitude exhibits simple complex pole terms, reflecting the presence ofthe pseudo bound state and the deuteron bound state, and a remainder which comprises theproper cut structure from multiple pion exchanges. To leading orders the free parameters of ourscheme are in one-to-one correspondence to s-wave scattering lengths and ranges and p-wavescattering volumes.

References[1] S. Weinberg; Nucl. Phys. B363 (1991) 3[2] U. van Kolck; in ”Low Energy Effective Theories and QCD”, D.-P. Min (ed.),

Seoul (1995)

22

Renormalisation of the SU(3) chiral meson-baryon lagrangian toorder q3

Guido MullerInstitut fur Theoretische Kernphysik

Nußallee 14-16D-53115 Bonn

e-mail:[email protected]

Three-flavor chiral perturbation theory with baryons is a topic of current interest. The dynamicsof kaon-nucleon interactions or kaon photo(electro)production are based on the SU(3) extensionof chiral effective lagrangians. We remind that for the isospin-odd pion-nucleon scattering lengththe loop correction only can fill the gap between the Weinberg-Tomozawa prediction and theempirical value. Loops produce in general ultraviolet divergences, which can be absorbed byintroducing counterterms [1]. The knowledge of the full divergence structure allows to controlthese calculations. We perform the complete regularisation of all Green functions with a singleincoming and outgoing baryon to order q3 in the chiral SU(3) meson-baryon system. Themethod is based on the work of Ecker who performed the complete renormalisation of Greenfunctions of the pion-nucleon interaction in the heavy baryon formalism [2]. This allows fora consistent chiral power counting. The divergences can be extracted in a chiral invariantmanner by making use of the heat kernel representation of the propagators in d-dimensionalEuclidean space. The main difference between the two calculations lies in the fact that thenucleons are in the fundamental representation of SU(2), while the baryons are in the adjointrepresentation of SU(3). This leads to some algebraic consequences for the construction of theone-loop generating functional [3].

References[1] J.Gasser, M.E. Sainio and A. Svarc, Nucl. Phys. B307 (1988) 779[2] G.Ecker, Phys. Lett. B336 (1994) 508[3] G.Muller and U.-G.Meißner, Bonn preprint, in preparation

23

Neutral Pion Photoproduction off Protons

Norbert KaiserTechnische Universitat Munchen, Physik Department T39

D-85747 Garching, James-Franck-Straße

I analyse the new threshold data for neutral pion photoproduction off protons in the thresholdregion [1,2] within the framework of heavy baryon chiral perturbation theory at order q4. Itis shown that indeed large loop corrections as predicted by CHPT at order q4 are needed tounderstand the small value of the S-wave multipole E0+ at threshold [3]. Due to the ratherslow convergence of this quantity in powers of the pion mass it does not provide anymore agood testing ground of chiral dynamics. However, there are new and rapidly converging lowenergy theorems for two combinations of the P-wave multipoles [3]. The one for P1 holds atthe few percent level when compared to the new TAPS and SAL data [1,2]. The low energytheorem for P2 can be tested soon with polarized photon data taken at MAMI. The few lowenergy constants entering the calculation can be well understood from resonance saturation [4].

Furthermore, I discuss double π0 photoproduction off protons close to threshold. The chiralloops enter here already at leading nonvanishing order and considerably enhance the nearthreshold total cross section. This feature remains in a full order q4 calculation of all next-to-leading order corrections [5].

References[1] M. Fuchs et al., Phys. Lett. B368 (1996) 20.[2] J. Bergstrom et al., Phys. Rev. C53 (1996) R1052.[3] V. Bernard, N. Kaiser and U. Meißner, Z. Phys. C70 (1996) 483.[4] V. Bernard, N. Kaiser and U. Meißner, ”Chiral Symmetry and the Reaction γp → π0p”,Phys. Lett. B (1996) in print.[5] V. Bernard, N. Kaiser and U. Meißner, ”Double Neutral Pion Photoproduction at Thresh-old”, Phys. Lett. B (1996) in print.

24

Photoproduction of Pions off the Nucleon - Results from DispersionTheory

Dieter Drechsel and Olaf HansteinInstitut fur Kernphysik, Universitat Mainz, D-55099 Mainz, Germany

Dispersion relations at constant t [1] have been used to analyze the recent precision experiments atMAMI (Mainz) and ELSA (Bonn). The partial wave amplitudes fulfil the phase relations required bythe Watson theorem at the lower energies and some approximate relation at the higher energies. Thecontributions of the dispersion integral above 2GeV are replaced by a fraction of the vector mesonexchange representing the Regge behaviour of the amplitudes expected at very high energies. Thisunknown high-energy behaviour and the necessity to add multipoles of the homogeneous equationsto the solutions of the coupled system of multipole amplitudes, leads to the introduction of 9 freeparameters determined by a fit to the data in the energy region of 160MeV ≤ Eγ ≤ 450MeV .

The threshold region has not been included in our fit because isospin symmetry breaking effects playan important role in that region due to the different pion masses. However, the threshold predictionsobtained from our analysis are in very good agreement with the results from chiral perturbation theory(ChPT ) [2]. In particular, we find E0+(nπ

+) = 28.4 ·10−3/mπ and E0+(pπ−) = −31.9 ·10−3/mπ. The

latter value is consistent with the angular distribution and the total cross section of a recent TRIUMFexperiment [3] whose analysis had led to a threshold amplitude of E0+(pπ

−) = −34.7 · 10−3/mπ, indisagreement with ChPT an d low energy theorems. It is also interesting that the data at the higherenergies, via dispersion relations, lead to a prediction of E0+(pπ

0) = −0.4 · 10−3/mπ at π+- threshold.This is in good agreement with both the data [4] and ChPT , and a consequence of the importance ofloop corrections to the ”old” low energy theorem.

In the region of the ∆(1232) isobar, we have decomposed the E(3/2)1+ and M

(3/2)1+ multipoles into

resonance and background contributions using the speed-plot technique [5]. In this way we are ableto determine the position of the resonance pole in the complex plane at W = MR − iΓR/2 withMR = (1211 ± 1)MeV and ΓR = (100 ± 2)MeV [6], in excellent agreement with results from pion-nucleon scattering. The resonant contributions to the two multipoles are then determined as thecomplex residues at the resonance pole, RE/Mexp(iφE/M ). While theWatson theorem requires that the

physical amplitudes E(3/2)1+ and M

(3/2)1+ have the same phase, the corresponding ratio for the resonant

amplitude is a complex number, REexp(iφE)/RMexp(iφM ) = (−.035,−0.46). Recent experimentson angular distributions and photon asymmetries [7] have found an E/M ratio of (−2.4 ± 0.2)% atW = 1232MeV . Since the experiment is sensitive to the ratio Re{E∗

1+M1+}/ | M1+ |2, our analysispredicts that the resonant contribution to the E/M ratio is −3.5%.

In conclusion, dispersion relations at constant t give a possibility to analyze and to interpret the newprecision data on pion photoproduction. Further improvements are expected to come from subtracteddispersion relations, by using the threshold results from ChPT as input, with the consequence ofreducing the sensitivity on the unknown behaviour of the dispersion integrals at the higher energies.References

[1] O. Hanstein, Ph. D. thesis, Mainz (1996).[2] V. Bernard et al., Nucl. Phys. B383 (1992) 442, and Z. Phys. C70 (1996) 483.[3] Kailin Liu, Ph. D. thesis, University of Kentucky (1994).[4] R. Beck et al., Phys. Rev. Lett. 65 (1990) 1841, and M. Fuchs et al., Phys. Lett. B368 (1996) 20.[5] G. Hoehler, πN Newsletter 7 (1992) 94 and 9 (1993) 1.[6] O. Hanstein, D. Drechsel and L. Tiator ”The position and the residues of the delta resonance polein pion photoproduction” (to be published).[7] R. Beck, Proc. Int. Conf. ”Baryons ’95”, Santa Fe (1995), and H.-P. Krahn, Ph. D. thesis, Mainz(1996).

25

Baryon Masses and σ–termsto second order in the quark masses

Bugra Borasoy

We analyze the octet baryon masses and the pion/kaon–nucleon σ–terms in the frameworkof heavy baryon chiral perturbation theory, [1]–[4]. We include all terms up-to-and-includingquadratic order in the light quark masses, mq. The pertinent low–energy constants are fixedfrom resonance exchange within the one–loop approximation. This includes contribution fromloop graphs with intermediate the spin–3/2 decuplet and the spin–1/2 octet states and fromtree graphs including scalar mesons. We demonstrate that two–loop corrections indeed modifythe leading one–loop results for some of these coefficients. Retaining only the contributions tothe low–energy constants to one–loop order, the only free parameter is the baryon mass in the

chiral limit,◦m. We find

◦m= 840 ± 100 MeV, [5],[6]. While the corrections of order m2

q aresmall for the nucleon and the Λ, they are still large for the Σ and the Ξ. Therefore a definitivestatement about the convergence of three–flavor baryon chiral perturbation can not yet bemade. The pion–nucleon σ–term is given parameter–free, we get σπN (0) = 43±10 MeV, whichis in good agreement with dispersion-theoretical determinations, together with the strangenesscontent of the nucleon, y = 0.08± 0.12. We also estimate the kaon–nucleon σ–terms, the shiftsto the respective Chang–Dashen points and some two–loops contributions to the nucleon mass.

References[1]E. Jenkins, Nucl. Phys. B368 (1992) 190[2]V. Bernard, N. Kaiser and Ulf-G. Meißner, Z. Phys. C60 (1993) 111[3]R.F. Lebed and M.A. Luty, Phys. Lett. B329 (1994) 479[4]M.K. Banerjee and J. Milana, Phys. Rev. D52 (1995) 6451[5]B. Borasoy and Ulf–G. Meißner, Phys. Lett. B365 (1996)285[6]B. Borasoy and Ulf–G. Meißner, Bonn preprint TK-96/14 (1996)

26

The Master Formula Approach in the Pion and Nucleon Sector

James V. SteeleSUNY Stony Brook

Stony Brook, NY USA

The master formula approach uses an on-shell chiral reduction scheme in order to make predic-tions for meson-nucleon dynamics [1]. It relates scattering amplitudes to measurable vacuumcorrelation functions hence allowing the inclusion of resonances. In particular, it has beenapplied to ππ scattering where a model independent verification of the rho dominance in thevector channel was shown [2].

In addition, a semi-classical expansion in h ∼ 1/f 2π may be done. This is equivalent to

a momentum expansion in the pion sector. Two solutions of the master formula result: onereproduces CHPT and the other allows for an additional relation between the divergences,reducing the number of parameters to two at one-loop [1,2].

The extension of MF to the nucleon sector was also reported on at this conference. The1/fπ expansion is no longer equivalent to CHPT since there are many mass scales involved.The π-N sigma term is found to be given by the Goldberger-Treiman discrepancy to tree level[3]. Using this one finds naturally that every time gA appears it is replaced by gπNN .

A comparison with all form factors and many scattering amplitudes calculated in the rela-tivistic formulation of chiral perturbation theory [4,5] shows additional differences. The scalarform factor and πN → ππN scattering amplitude have additional momentum dependent termsproportional to the π-N sigma term at one-loop. Further analysis is in progress [6].

References[1] I. Zahed and H. Yamagishi, Ann. Phys. 246, 3 (96).[2] J.V. Steele, I. Zahed, and H. Yamagishi, hep-ph/9505330[3] J.V. Steele, I. Zahed, and H. Yamagishi, hep-ph/9512233[4] J. Gasser, M.E. Sainio, and A. Svarc NPB307 (88) 779.[5] V. Bernard, N. Kaiser, T.S.H. Lee, and Ulf-G. Meißner, Phys. Rep. 246 (94) 315; V.Bernard, N. Kaiser, and Ulf-G. Meißner, hep-ph/9507418.[6] J.V. Steele, I. Zahed, and H. Yamagishi, in preparation.

27




Scalar susceptibility and critical behavior in QCD and Schwingermodel

Andrei Smilgaa) and Jack Verbaarschot(b)

a) ITEP, Moscow, Russiab) SUNY at Stony Brook, Stony Brook, USA

We evaluate the leading infrared behavior of the scalar susceptibility

χ =∫d4x〈

Nf∑

i=1

qiqi(x)Nf∑

i=1

qiqi(0)〉 − V 〈Nf∑

i=1

qiqi〉2 =1

V∂2m logZ

∣∣∣m=0

, (1)

in QCD and in the multiflavor Schwinger model for small non-zero fermion mass m and/orsmall nonzero temperature as well as the scalar susceptibility for QCD at finite volume. InQCD, it is determined by one-loop chiral perturbation theory, with the result that the leadinginfrared singularity behaves as ∼ logm at zero temperature:

χIR =N2

f − 1

8π2

(Σ

F 2π

)2

logΛ2

M2π

, (2)

and as ∼ T/√m at finite temperature:

χIRT =

(N2f − 1)

4π

T√2m

(Σ

F 2π

)3/2 (1 +

1

8Nf

T 2

F 2π

). (3)

(where also two loop chiral graphs are taken into account). These are exact results to be checkedin lattice and/or instanton model numerical calculations.

In the Schwinger model with several flavors we use exact results for the scalar correlationfunction. We find that the Schwinger model has a phase transition at T = 0 with criticalexponents that satisfy the standard scaling relations and do not coincide with the mean fieldtheory predictions. The singular behavior of this model depends on the number of flavors witha scalar susceptibility that behaves as ∼ m−2/(Nf+1). At finite volume V we show that thescalar susceptibility is proportional to 1/m2V . Recent lattice calculations of this quantity byKarsch and Laermann [1] and the related lattice work by Kocic and Kogut [2] are discussed.

References[1] F. Karsch and E. Laermann, Phys. Rev. D50 (1994) 6954.[2] A. Kocic and J. Kogut, Phys. Rev. Lett. 74 (1995) 3109

28

Finite Volume Analysis of Chiral Symmetry Breaking in QCD

Jan SternDivision de Physique Theorique1, Institut de Physique Nucleaire


It is argued [1] that in QCD, there exists a natural possibility of spontaneous breakdown ofchiral symmetry (SBCHS) without quark-antiquark condensation in the vacuum. In contrastto the Nambu Jona-Lassinio model, the ground state of QCD might be characterized by a nonGaussian distribution of small eigenvalues of the (Euclidean) Dirac operator. This could lead tothe large volume behaviour V −1/2 of the averaged lowest Dirac levels. The corresponding leveldensity would not be sufficient to trigger qq condensation [2,3] ( the large volume behaviourV −1 is necessary ), although it could be sufficient to make appear Goldstone bosons coupled tothe conserved axial-vector currents. New sum rules for inverse powers of the Dirac eigenvaluesare derived [3,1] which could be suitable for a numerical study of the mechanism of SBCHS notsuffering from the usual drawbacks of lattice simulation near the chiral limit.

References[1] J. Stern, in preparation.[2] T. Banks and A. Casher, Nucl. Phys. B168 (1980) 103.[3] H. Leutwyler and A. V. Smilga, Phys. Rev. D46 (1992) 5607.


29

Elastic ππ scattering to two loops

Gilberto ColangeloInstitut fur Theoretische Physik

Bern, Switzerland

I have presented the calculation of the ππ scattering amplitude to two loops in ChiralPerturbation Theory. The calculation has been done without any approximations and theresult is given in analytical form [1].

A thorough numerical analysis is still in progress. However, if we use the current values forthe O(p4) constants, and estimate with resonance saturation the new O(p6) constants, we getsmall corrections to the quantities of direct experimental interest, like a00, a

00 − a20 and δ00 − δ11.

As an example, a00, which at one loop is 0.201 [2], becomes 0.217. The two loop calculationconfirms that CHPT can yield very sharp predictions for ππ scattering at low energy, as stressedin [2]. A clear discrepancy with experimental data would then require a significant revision ofour picture of the vacuum structure of QCD. As shown in [3], a value of a00 much larger thanthe CHPT predictions would be the signal for a quark–antiquark condensate much smaller thanwhat is usually assumed.

Finally I have compared the CHPT predictions to a recent lattice calculation of the twoS–wave scattering lengths [4]. Despite the systematic effects, such as quenching, the agreementis quite impressive. It would be interesting to improve the study of these scattering lengths onthe lattice to clarify whether the agreement is accidental or not.

References[1] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M. Sainio, Phys. Lett. B374 (1996) 210.[2] J. Gasser and H. Leutwyler, Phys. Lett. 125B (1983) 325.[3] M. Knecht, B. Moussallam, J. Stern and N.H. Fuchs, Nucl. Phys. B457 (1995) 513.[4] M. Fukugita, Y. Kuramashi, M. Okawa, H. Mino and A. Ukawa, Phys. Rev. D52 (1995)3003.

30

Low Energy ππ Scattering to Two Loops

Marc KnechtDivision de Physique Theorique1, Institut de Physique Nucleaire


The amplitude for elastic ππ scattering at low energy has been computed to two-loop accuracyin the chiral expansion [1]. As shown previously [2], it is determined by the Goldstone natureof the pion, combined with the general S-matrix properties of analyticity, crossing symmetryand unitarity, up to six independent combinations of low energy constants, denoted by α, β, λi,i = 1, 2, 3, 4, and which are not fixed by chiral symmetry. The four constants λi were determinedvia sum rules, evaluated using available data on ππ interaction at medium and high energies[3]. The remaining two parameters, α and β, have to be determined from low energy ππ data.An experimental determination of α is of particular importance. The value of this parameter isintimately correlated to the ratio x = −2m〈0|qq|0〉/F 2

πM2π , or equivalently, to the quark mass

ratio ms/m. The commonly accepted picture that spontaneous breakdown of chiral symmetryresults from a strong condensation of q − q pairs in the QCD vacuum requires that x ∼ 1(or r ∼ 25), and it is only compatible with values of α and β close to unity, as predicted bystandard χPT [4,5]. Unfortunately, a fit to the presently available ππ data obtained from Kℓ4

decays gives α = 2.16± 0.86, β = 1.074± 0.053, and remains thus inconclusive in this respect.Clearly, additional information, which would make such fits more accurate, is needed. Thismay be provided by new high statistics Kℓ4 experiments, from e. g. KLOE at DAΦNE, or bya precise measurement of the lifetime of π+π− atoms, as planed by the DIRAC experiment atCERN.

On the other hand, varying α and β in the above ranges leads to values of the thresholdparameters which are in perfect agreement with the results obtained from Roy equation analysesof available Kℓ4 data. In the range of energies accessible in Kℓ4 decays, the two-loop chiralexpression of the ππ amplitude together with the determination of the four constants λi thuscontains all the relevant information on the ππ interaction which is already encoded in the Royequation.

References[1] M. Knecht, B. Moussallam, J. Stern and N. H. Fuchs, Nucl. Phys. B457 (1995) 513.[2] J. Stern, H. Sazdjian and N. H. Fuchs, Phys. Rev. D47 (1993) 3814.[3] M. Knecht, B. Moussallam, J. Stern and N. H. Fuchs, Nucl. Phys. B, in print.[4] J. Gasser and H. Leutwyler, Phys. Lett. B125 (1983) 325; Ann. Phys. 158 (1984) 142.[5] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M. Sainio, Phys. Lett. B374 (1996) 210.


31

Constraints on π-π scattering threshold parameters from low energysum rules

B. Ananthanarayan, D. Toublan and G. WandersInstitut de Physique Theorique, Universite de Lausanne,

CH 1015, Lausanne, Switzerland

In ChPT, the effective theory of QCD at low energy, the π-π scattering threshold parametersplay a central role [1,2]. Despite the available phase shift analysis of the S- and P -waves, theexperimental information on the threshold parameters is not very accurate [3,4].

π-π scattering being a fundamental strong interaction process well suited for theoreticalinvestigations, the principles of axiomatic field theory lead to a wealth of rigorous results [5].Using analyticity, unitarity and crossing symmetry, and with the help of the homogeneousvariables [6] three sum rules involving dispersion integrals dominated by low-energy S- and P -waves can be constructed from amplitudes which are completely symmetric in the Mandelstamvariables [7].

The dispersion integrals depend significantly on poorly known threshold parameters.Thislead us to a parametrization of the S- and P -wave absorptive parts occuring in the integrandsreproducing the main features of the cross sections above threshold whereas the scatteringlengths and effective ranges remain free parameters [8]. The sum rules are turned into nonlinearequations for the S- and P -wave threshold parameters and a combination of D-wave scatteringlengths. We show that the solutions of these equations which are compatible with the data areconfined to a rather small portion of the experimentally allowed domain and enforce a strongcorrelation between them [7]. This is our main result and it establishes the relevance of our sumrules. Furthermore both Standard and Generalized ChPT satisfy these sum rules constraintsat the one-loop level.

References[1] J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.)158, 142 (1984).[2] J. Stern, H. Sazdjian and N. H. Fuchs, Phys. Rev. D47, 3814 (1993).[3] B. R. Martin, D. Morgan and G. Shaw, “Pion-Pion Interaction in Particle Physics,” Aca-demic Press, London/New York, 1976; also see W. Ochs, π N News letter, 3, 25 (1991).[4] M. M. Nagels, et al., Nucl. Phys. B147, 189 (1979).[5] A. Martin, “Scattering Theory: Unitarity, Analyticity and Crossing,” Springer-Verlag,Berlin, Heidelberg, New York, 1969.[6] G. Wanders, Helv. Phys. Acta. 39, 228 (1966).[7] B. Ananthanarayan, D. Toublan and G. Wanders, Phys. Rev. D53, 2362 (1996).[8] A. Schenk, Nucl. Phys. B363, 97 (1991).

32

Heavy Quark Solitons

Herbert Weigel

Institute for Theoretical Physics, Tubingen University

Auf der Morgenstelle 14, D-72076 Tubingen, Germany

A generalization of the effective meson Lagrangian possessing the heavy quark symmetry(HQS) [1] to finite meson masses is employed to study the meson mass dependence of the

spectrum of S– and P wave baryons containing one heavy quark or anti–quark. These baryons

are described as respectively heavy mesons or anti–mesons bound in the background of a solitonof the light meson fields [2]. No further approximation is made to solve the bound state

equations for S– and P wave heavy baryons. It is observed that the HQS–prediction for thebinding energies of these baryons is approached only very slowly as the mass M of the heavy

meson increases [3]. On the other hand the bound state wave–functions satisfy the HQS–relations reasonably well at masses as low as M ≈ 5GeV [3].

For physically relevant mass parameters associated with the charm and bottom sectors, twotypes of models supporting soliton solutions for the light mesons are considered: the Skyrme

model of pseudoscalars only as well as an extension containing also light vector mesons. It hasbeen found that only the Skyrme model with vector mesons provides a reasonable description

of the spectrum of both light and heavy baryons [3]. It furthermore turns out that the anti–quarks are unbound in the charm sector and only weakly bound, if at all, in the bottom

sector [3]. Subsequently the system consisting of the vector meson soliton and the heavymeson bound state is projected onto states with good spin and isospin. As consistency check

it has been shown that the mass gap between heavy baryons with spin 12and 3

2decreases

as 1/M . Turning again to the physical parameters the model predicts the following massdifferences: M(ΣC)−M(ΛC) = 178MeV, M(ΛC)−M(N) = 1.321GeV and M(ΛB)−M(N) =

4.495GeV. These compare reasonably well with the empirical values 165MeV, 1.345GeV and(4.701± 0.050)GeV, respectively.

In the heavy quark limit the coupling between mesons containing a heavy quark and thesoliton of the Nambu–Jona–Lasinio model is studied in addition. As this soliton configuration

contains quark fields with non–vanishing grand spin conceptually different coupling schemesbetween the heavy meson field and the soliton are discovered [4]. These new schemes appear in

addition to those which are already present [2,3] in Skyrme type models and may yield a largerbinding of the baryon with a heavy quark [4].

References[1] M. Neubert, Phys. Rep. 245 (1994) 259.

[2] C. Callan and I. Klebanov, Nucl. Phys. B262 (1985) 365;D. P. Min, Y.Oh, B. Park, and M. Rho, Int. J. Mod. Phys. E4 (1995) 47.

[3] J. Schechter, A. Subbaraman, S. Vaidya, and H. Weigel, Nucl. Phys. A590 (1995) 655; E:

Nucl. Phys. A598, 583 (1996).[4]L. Gamberg, H. Weigel, U. Zuckert, and H. Reinhardt, Heavy Quark Solitons in the Nambu–

Jona-Lasinio Model, hep–ph/9512294.

33

http://arxiv.org/abs/hep--ph/9512294

Low Energy QCD in the large Nc Limit

Eduardo de Rafael

Centre de Physique TheoriqueCNRS-Luminy, Case 907

F-13288 Marseille Cedex 9, FRANCE.

My talk starts with a quick historical review of developments of QCD in large Nc. I also

compare the observed hadronic spectrum with the one expected from large Nc. I use this asa motivation to introduce and discuss the ENJL–model (see Ref.[1] and references therein,)

as a low energy model of QCD at large Nc . The successes and drawbacks of this model arereviewed.

I emphasize the need to develop good models for the low energy QCD effective action. Thisis most dramatic in the non–leptonic sector of the Standard Model. I show examples of low–

energy observables which require the knowledge of euclidean Green’s functions at all values ofthe euclidean momenta: the hadronic contributions to the muon g−2; and the electromagnetic

π+ − π0 mass difference are two examples I discuss in the light of the ENJL–model. I showhow the problem of matching long– and short–distances can be successfully solved in these two

cases. I also discuss how the combined hypothesis: large Nc and 〈ψψ〉 → 0, require at leasttwo more Weinberg sum rules which implies severe phenomenological constraints (see Ref.[2].)

I next discuss, within the example of the Adler’s function, the general question of matching

short distance QCD behavior obtained within the QCD sum rules a la SVZ, with the longdistance hadronic behavior as predicted by the ENJL–model. (See Ref.[3].) This clearly shows

no overlap of the two regimes, and the need for a GV 6= 0. I then discuss the possibility ofmatching the two regimes using the QCD–Hadronic Duality arguments developed in Ref.[4],

and suggest a similar approach to solve the problem of matching encountered in the recentcalculations of the light–by–light hadronic contributions to the muon g − 2 and the BK–factor

discussed in this Workshop, (see these proceedings.)In the last part of my talk I review some recent exact results for low energy observables

which, following Ref.[5], have been obtained within the framework of a simultaneous expansionin the large Nc limit and U(3)L × U(3)R chiral perturbation theory. See in particular Refs.[6]

and [7].References

[1] J. Bijnens, Chiral Lagrangians and Nambu–Jona-Lasinio like Models, Phys. Reports 265,No6 (1996), and references therein.

[2] J. Bijnens and E. de Rafael, in progress.

[3] S. Peris, M. Perrottet and E. de Rafael, in progress.

[4] R.A. Bertlmann, G. Launer and E. de Rafael, Nucl. Phys. B250, 61 (1985).

[5] S. Peris and E. de Rafael, Phys. Letters, 348B, 539 (1995).[6] S. Peris, M. Perrottet and E. de Rafael, Phys. Letters 355B, 523 (1995).

[7] A. Pich and E. de Rafael, hep-ph/9511465, to appear in Phys. Letters.

34


Dispersive Analysis of the Predictions of

Chiral Perturbation Theory for ππ Scattering

M.R. Pennington 1 and J. Portoles 2

1 Centre for Particle Theory, University of Durham, Durham DH1 3LE, U.K.2 INFN, Sezione di Napoli, I-80125 Naples, Italy

A way of testing the ππ predictions of Chiral Perturbation Theory against experimental datais to use dispersion relations to continue experimental information into the subthreshold region

where the theory should unambiguously apply. Chell and Olsson [1] have proposed a test ofthe subthreshold behaviour of chiral expansions which highlights potential differences between

the Standard [2] and the Generalized [3] forms of the theory. We illustrate how, with currentexperimental uncertainties, data cannot distinguish between these particular discriminatory

coefficients despite their sensitivity [4]. Nevertheless, the Chell-Olsson test does provide aconsistency check of the chiral expansion, requiring that the O(p6) corrections to the discrim-

inatory coefficients in the Standard theory must be ∼ 100%. Indeed, some of these have beendeduced [5] from the new O(p6) computations [6] and found to give such large corrections. One

can then check that the O(p8) corrections must be much smaller.

We conclude that this test, like others, cannot distinguish between the different forms of ChiralSymmetry Breaking embodied in the alternative versions of Chiral Perturbation Theory with-

out much more precise experimental information near threshold.

References[1] E. Chell, Ph.D. thesis submitted to the University of Wisconsin;

M.G. Olsson, Chiral Dynamics: Theory and Experiment, eds. A.M. Bernstein,B.R. Holstein, (Springer-Verlag, 1995) pp. 111-112.

[2] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142; Nucl. Phys.B250 (1985) 465.

[3] J. Stern, H. Sazdjian and N.H. Fuchs, Phys. Rev. D47 (1993) 3814.

[4] M.R. Pennington and J. Portoles, submitted to Physical Review D.[5] B. Moussallam, private communication.

[6] M. Knecht, B. Moussallam, J. Stern and N.H. Fuchs, Nucl. Phys. B457 (1995) 513; J.Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. Sainio, Phys. Lett. B374 (1996) 210.

35

FSI in η → 3π and the quark mass ratio Q2

Christian Wiesendanger

Dublin Institute for Advanced Studies, School of Theoretical Physics10 Burlington Road, Dublin 4, Ireland

To leading order the mass ratios of the three light quark flavours u, d, s are easily accessi-ble and known for a long time. The next-to-leading order analysis has been performed by Gasser

and Leutwyler [1]. They have shown that the quantityQ2 = m2s−m2

m2d−m2

u=

M2K

M2π

M2K−M2

π

M2

K0−M2

K+

(1 +O(m2))

is given by the above ratio of pure QCD meson masses, up to corrections of second order. To use

the experimental mass values for the mesons one has to correct for the e.m. mass contributions.This is highly controversial as Dashen’s theorem may receive large corrections [2].

An independent way to measure Q2 is provided by the isospin-violating decay η → 3π as

the corresponding rate is proportional to Q−4 [3]. Sutherland’s theorem proves to be stable [4]and the main uncertainties in obtaining a reliable rate come from the strong FSI of the π’s. To

evaluate those Kambor, Wiesendanger and Wyler [5] use extended Khuri-Treiman equations.The subtraction to the dispersion relation may then be fixed by the one-loop amplitude of

Gasser and Leutwyler [3]. The FSI corrections are moderate and enhance the amplitude by14% at the center of the Dalitz plot. This reduces the usual value for Q2 = 24.1 obtained with

Dashen to Q2 = 22.4 ± 0.9. In agreement with this result Anisovich and Leutwyler [6] haveobtained Q2 = 22.7± 0.8 in their dispersive analysis.

References[1] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465.

[2] J. Donoghue, B. Holstein and D. Wyler, Phys. Rev. D47 (1993) 2089;R. Baur and R. Urech, Phys. Rev. D53 (1996) 6552;

J. Bijnens, Phys. Lett. B306 (1993) 343.[3] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 539.

[4] R. Baur, J. Kambor and D. Wyler, Nucl. Phys. B460 (1996) 127.

[5] J. Kambor, C. Wiesendanger and D. Wyler, Nucl. Phys. B465 (1996) 215.[6] A.V. Anisovich and H. Leutwyler, Dispersive analysis of the decay η → 3π, hep-ph/9601237.

36


Heavy Baryon ChPT with Light DeltasThomas R. Hemmert

Department of Physics and Astronomy, University of Massachusetts, Amherst, MA 01003 USA

In recent years baryon chiral perturbation theory has matured into a systematic field theory. Startingfrom work in the fully relativistic framework [1], the introduction of the heavy mass formalism [2] led tothe development of the so called Heavy Baryon Chiral Perturbation Theory (HBChPT) which alloweda consistent chiral power counting [3] to all orders. Once one goes beyond the leading order lagrangianin the baryon sector, one encounters two different classes of vertices: One class is accompanied byunknown counterterms analogous to the the meson sector whereas the other class corresponds to theso called 1/m corrections (relativistic corrections) to vertices of lower order in the chiral expansion.Both classes of vertices and the interplay between them are now well understood for the case of spin1/2 nucleons up to chiral order O(p3) [4].

In the scheme of HBChPT described so far, all baryon resonances are treated as being infinitelyheavy and decoupled from the theory [5]. Therefore they only contribute to higher order countert-erms in the chiral lagrangian through effective contact interactions. However, it is well known fromphenomenology that the first nucleon resonance ,∆(1232), plays a strong role in low energy baryonprocesses. It has therefore been advocated for quite a while [6] that one should keep the lowest lyingspin 3/2 baryon resonances as explicit degrees of freedom in the chiral lagrangian. Several calculationsalong this line of thinking exist in the literature. However, many of these calculations are incomplete.In particular, the construction of the above mentioned 1/m corrected vertices involving spin 3/2 fieldshas been missing in the literature.

We report on recent work [5,7] in SU(2) HBChPT that allows a systematic treatment of spin 1/2nucleon and explicit ∆(1232) degrees of freedom. Following the approach of [3], we start from themost general relativistic spin 3/2 lagrangian, explicitly keeping ”point-transformation” invariance andall possible ”off-shell” coupling structures. After having separated the spin 3/2 and the spurious spin1/2 components of the Rarita-Schwinger spinor via a projector formalism, we make the transitionto the heavy mass formalism. To leading order, we reproduce the results of [3] (NN-sector) and [6](∆∆,∆N -sector). In next-to-leading order [O(p2)], we explicitly construct all 1/m corrected verticesfor the NN , N∆ and ∆∆ lagrangians. We also discuss how the O(p2) NN lagrangian of [3] has tobe changed, once one allows for explicit ∆(1232) degress of freedom in the theory. This leads us to anew understanding of ”resonance saturation” in the baryon sector (see [5] for details). Furthermore,we discuss the O(p2) vertices of the ∆∆ and N∆ lagrangians accompanied by counterterms and showhow our methods can be generalised to obtain the corresponding lagrangians beyond O(p2).

As a specific example we discuss the effect of ∆(1232) in the process of π0 photoproductionat threshold. Keeping the delta-resonance in the theory introduces a new mass scale ∆ = M∆ −MN ≈ 300MeV, which is non-vanishing in the chiral limit, nevertheless small compared with thechiral symmetry breaking scale Λχ ≈ 1GeV. We therefore organise the calculation into a δ-expansion,where δ corresponds to any of the small quantities p,mπ,∆. Calculating up to order δ3, we find thatthe leading order contribution is given by a diagram involving one of the new O(p2) 1/m correctedN∆π-vertices. We compare this result with a standard HBChPT calculation [8] that has the deltas”frozen out” and close with a numerical estimate.[1] J. Gasser, M.E. Sainio, A. Svarc; Nucl. Phys. B307 (1988) 779[2] E. Jenkins, A. Manohar; Phys. Lett. B255 (1991) 558[3] V. Bernard et al.; Nucl. Phys. B388 (1992) 315[4] G. Ecker; Phys. Lett. B336 (1994) 508[5] J. Kambor, these proceedings[6] E. Jenkins, A. Manohar; Phys. Lett. B259 (1991) 353[7] T.R. Hemmert, B.R. Holstein, J. Kambor; forthcoming[8] V. Bernard, N. Kaiser, U.G. Meissner; Z.Phys. C70 (1996) 483

37

Resonance Saturation in the Baryonic Sector of Chiral PerturbationTheory

Joachim KamborDivision de Physique Theorique, Institut de Physique Nucleaire

F-91406 Orsay Cedex

Heavy baryon chiral perturbation theory (HBChPT) [1,2] including spin 3/2 delta-resonancedegrees of freedom [3] has recently been reformulated by making use of a 1/m-expansion, mbeeing the nucleon mass [4,5,6]. The theory admits a systematic expansion in the small scale δ,where δ collectively denotes soft momenta, the pion mass or the delta-nucleon mass difference.It is pointed out that valuable information about HBChPT can be obtained by comparing thechiral expansion with the δ-expansion. Large corrections originating from intermediate deltascan be identified and the convergence of the chiral expansion with respect to these effects canbe studied. Renormalization as well as the different meaning of counterterms in HBChPT andthe δ-expansion, respectively, is discussed in detail. This leads directly to a reformulation ofresonance saturation in the baryonic sector of ChPT [6]. As an explicit example, the scalarsector of one-nucleon processes in chiral SU(2) is worked through. In particular, it is shownthat the shift of the scalar form factor of the nucleon between the Cheng-Dashen point andzero, σ(2m2

π)− σ(0) ≈ 15 MeV [7], has a natural explanation in the δ-expansion.

References

[1] J. Gasser, M.E. Sainio, and A. Svarc, Nucl. Phys. B307 (1988) 779.[2] V. Bernard et al., Nucl. Phys. B388 (1992) 315.[3] E. Jenkins and A.V. Manohar, Phys. Lett. B259 (1991) 353.[4] T.R. Hemmert, these proceedings.[5] J. Kambor, Heavy Baryon Chiral Perturbation Theory and the Spin 3/2 Delta Resonances,talk given at the 7th International Conference on the Structure of Baryons, Santa Fe, NM, 3-7Oct 1995, Orsay preprint IPNO/TH 96-13.[6] T.R. Hemmert, B.R. Holstein, and J. Kambor, in preparation.[7] J. Gasser, H. Leutwyler, and M.E. Sainio, Phys. Lett. B253 (1991) 252, 260.

38

Novel Algebraic Consequences of Chiral Symmetry

Silas R. BeaneDuke University

Durham, NC 27708-0305USA

The empirical success of the Ademollo-Veneziano-Weinberg mass relation provides an ex-ample of regularity in the hadronic spectrum that remains unexplained by the symmetries ofQCD [1]. We provide an explanation for the success of this relation based on the premise thatall hadrons fill out —in general— reducible representations of SU(2)×SU(2) [2]. Mass-squaredmatrix elements of heavy hadrons and light hadrons are related using heavy quark and chiralsymmetries [3]. Our result suggests that hadrons might be profitably viewed as bound states ofweakly interacting, parity-doubled constituent quarks. We illustrate the essence of our resultusing a simple effective lagrangian model.

References[1] M. Ademollo, G. Veneziano, and S. Weinberg, Phys. Rev. Lett. 28 (1968) 83.[2] S. Weinberg, Phys. Rev. Lett. 65 (1990) 1177; ibid, 1181.[3] S.R. Beane, DUKE-TH-95-98, hep-ph/9521228.

39


Hadronic contributions to the muon g-2: an updated analysisElisabetta Pallante, Johan Bijnens and Joaquım Prades.

NORDITA, Blegdamsvej 17, DK-2100, Copenhagen, Denmark

The anomalous magnetic moment of the muon is one of the best candidates to probe the electroweaksector of the Standard Model. For a review of the theoretical aspects see e.g. [1]. Three typesof contributions are present: the pure QED contributions, the hadronic contributions and the weakcontributions. Pure QED contributions are largely dominant aQED

µ = 11658 470.6(0.2) · 10−10 [1],

but the leading hadronic vacuum polarization contribution is sizable ah.v.p.µ = 725.04(15.76) · 10−10

[2] and theoretically predicted with an uncertainty of the same size of the weak contributions aEWµ =15.1(0.4) · 10−10 [3]. To disentangle weak contributions we need to further reduce the theoreticaluncertainty which affects the hadronic sector. A new BNL experiment is planning to reach an accuracyof ±40 · 10−11 in the determination of the muon aµ = (g − 2)/2, more than a factor of twenty ofreduction respect to the latest determination at the Cern Storage Ring aexpµ = 11659 230(84) · 10−10.This motivated the recently raised interest in the theoretical determination of this observable withan improved accuracy. At present hadronic contributions are the main source of uncertainty in thetheoretical prediction. We distinguish three classes of hadronic contributions to aµ: a) hadronicvacuum polarization (h.v.p.) contribution which appears at order (α/π)2 b) higher order correctionsto the hadronic vacuum polarization diagram [4] and c) light-by-light scattering contributions whichstart at order (α/π)3.

The h.v.p. contribution can be extracted via phenomenological dispersive analysis from the totale+e− → hadrons cross section. This is at present the most accurate way of determination [2]. Tofurther reduce its uncertainty new more precise data of e+e− → hadrons cross section are needed.Alternatively a low energy effective model of QCD can be used. In spite of the lack of confinementand the theoretical debated connection with QCD the Extended Nambu-Jona Lasinio (ENJL) model[5] does satisfy few phenomenological constraints (e.g. Weinberg Sum Rules) which are necessaryconditions to guarantee a good matching with PQCD and provides a systematic treatment of observ-ables dominated by long distance dynamics. Its prediction of the h.v.p. contribution [6] is in goodagreement with phenomenological determinations.

A novel determination of the light-by-light scattering contribution has been proposed in [7] withinthe ENJL framework (see also [8] for an alternative derivation). The dominant contribution isthe twice anomalous pseudoscalar exchange diagram. The final result we get is alight−by−light

µ =(−9.2 ± 3.2) · 10−10. This is between two and three times the expected experimental uncertainty atthe forthcoming BNL muon g − 2 experiment. Adding the other Standard Model contributions to aµthe present theoretical estimate for the muon g − 2 is athµ = 11659 182(16) · 10−10.[1] “Frontiers of High Energy Spin Physics”,Nagoya 1992, T. Hasegawa et al. (eds.), (Universal Acad.Press, Tokyo, 1992); “The Future of Muon Physics”, Heidelberg, Germany (1991), Z. Phys. C56, K.Jungmann, V.W. Hughes, and G. zu Putliz (eds.); “Quantum Electrodynamics”, T. Kinoshita (ed.),World Scientific, Singapore, (1990).[2] S. Eidelman and F. Jegerlehner, Z. Phys. C67 (1995) 585 .[3] A. Czarnecki, B. Krause, and W.J. Marciano, Phys. Rev. D52(1995)2619, Karlsruhe preprintTTP95-34(1995), hep-ph/9512369; S. Peris, M. Perrottet, and E. de Rafael, Phys. Let. B355(1995)523.[4] T. Kinoshita, B. Nizic, and Y. Okamoto, Phys. Rev. D31 (1985) 2108.[5] J. Bijnens, C. Bruno and E. de Rafael, Nucl. Phys. B390(1993)501; J. Bijnens, Phys. Rep.265(1996)369;[6] E. de Rafael, Phys. Lett. B322 (1994) 239; E. Pallante, Phys. Lett. B341 (1994) 221.[7] J. Bijnens, E. Pallante, and J. Prades, Phys. Rev. Lett. 75 (1995) 1447; Erratum: ibid. 75 (1995)3781; hep-ph/9511388, to appear in Nucl. Phys. B.[8] M. Hayakawa, T. Kinoshita, and A.I. Sanda, Phys. Rev. Lett. 75 (1995) 790; Nagoya Univ.preprint DPNU-95-30 (1995).

40



Some Hadronic Matrix Elements within the Extended NJL Model

Johan Bijnensa) and Joaquim Pradesb)

aNORDITA, Blegdamsvej 17, DK-2100 Copenhagen (Denmark).

bDepartament de Fısica Teorica, Universitat de Valencia and IFIC, Universitat deValencia-CSIC

C/ del Dr. Moliner 50, E-46100 Burjassot (Valencia) Spain.

The test of the Standard Model at low energy is generally affected by the uncertaintyassociated with the calculation of hadronic matrix elements at low energy. Even at low energies,the calculation of hadronic matrix elements requires, in general, the knowledge of the stronginteracions at all scales. This is for instance what happens in the BK parameter [1] or in thecorrections to the Dashen’s theorem [2]. There, a virtual boson (W s or photon) is integratedout making the internal scale to run from zero up toMW (∞). There are other hadronic matrixelements (like γγ → π0π0) [3] that start at high order within CHPT and therefore are moresensible to the high energy behaviour of QCD. Also in this type of hadronic matrix elementsone would like to obtain some matching with QCD. We have atacked the problem of calculatinghadronic matrix elements using the Extended NJL model version in Refs. [4,5,6,7] as a goodhadronic model at low energies and imposing short distance QCD behaviour at high energies.Though the matching obtained is not very good and more work to improve the intermediateenergy region is needed, we have already obtained interesting results [1,2,3]. Work in thesame direction is in progress for ∆S = 1 decays like K → π, 2π. For some work to improve thematching between the low-energy contributions and the shorty distance for two point functions,see the contribution by Eduardo de Rafael to this Workshop and references therein.

References[1] J. Bijnens and J. Prades, Nucl. Phys. B444 (1995) 523.[2] J. Bijnens and J. Prades, in preparation.[3] J. Bijnens, A. Fayyazuddin, and. J. Prades, preprint FTUV/95-70 (1995) (to be publishedin Phys. Lett. B).[4] J. Bijnens, C. Bruno, and E. de Rafael, Nucl. Phys. B390 (1993) 501.[5] J. Bijnens, E. de Rafael, and H. Zheng Z. Phys. C62 (1994) 437.[6] J. Bijnens and J. Prades, Z. Phys. C64 (1994) 475.[7] J. Bijnens, Phys. Rep. 265 (1996) 369.

41

On the Corrections to Dashen’s Theorem

Res UrechInstitut fur Theoretische Teilchenphysik, Universitat Karlsruhe

D-76128 Karlsruhe, Germany

The electromagnetic corrections to the masses of the pseudoscalar mesons π and K are con-sidered. At order O(e2) in the chiral limit Dashen’s theorem [1] is given by the relation∆M2

K − ∆M2π = 0, where ∆M2

P = M2P± − M2

P 0 . At order O(e2mq) this relation is subjectto corrections, which are probably large [2]. We calculate the contributions at order O(e2mq)that arise from resonances within a photon loop in the framework of chiral perturbation theory[3]. Within this approach we find rather moderate deviations to Dashen’s theorem [4].

References

[1] R.Dashen, Phys. Rev. 183 (1969) 1245.[2] K.Maltman and D.Kotchan, Mod. Phys. Lett. A5 (1990) 2457;

J.F.Donoghue, B.R.Holstein and D.Wyler, Phys. Rev. D47 (1993) 2089;J.Bijnens, Phys. Lett. B306 (1993) 343;R.Urech, Nucl. Phys. B433 (1995) 234;H.Neufeld and H.Rupertsberger, Z. Phys. C68 (1995) 91.

[3] G.Ecker, J.Gasser, A.Pich and E.de Rafael, Nucl. Phys. B321 (1989) 311.[4] R.Baur and R.Urech, Phys. Rev. D53 (1996) 6552.

42

K → πγγ decays : unitarity corrections

and vector meson contributions

G. D’Ambrosio and J. Portoles

INFN, Sezione di Napoli, I-80125 Naples, Italy

K → πγγ are interesting processes by themselves as ChPT tests and KL → π◦γγ in particularmight have an important role as a CP conserving amplitude contributing to KL → π◦e+e−.

Two different helicity amplitudes contribute to KL → π◦γγ : A and B. The first appearsat O(p4) [1], it is vanishing for small diphoton invariant mass and generates a suppressed am-plitude for KL → π◦e+e−. The second amplitude B appears at O(p6), it is non–vanishing forsmall diphoton invariant mass and generates an unsuppressed amplitude for KL → π◦e+e− [2].Though the experimental spectrum for KL → π◦γγ seems very well reproduced by the O(p4)leading contribution the rate is not. This has lead several authors to consider some O(p6) con-tributions (see references quoted in [2]) : i) unitarity corrections from physical KL → π◦π+π−

amplitude give a 20 − 30% increase in the amplitude with a slight deformation of the O(p4)spectrum, ii) an appropriate choice of the B amplitude generated by vector meson contributionscan accomodate width and spectrum, and also iii) unitarization of the ππ intermediate statesamplitude with inclusion of the experimental γγ → π◦π◦ amplitude should help.

K+ → π+γγ is also an appealing channel which will be measured soon. The leadingcontribution is O(p4) with loops and local contributions which size is an interesting test ofweak hadron dynamics [3].

We show [4] that unitarity corrections to K+ → π+γγ are important and generate also a20−30% increase in the B amplitude. We then study [5] O(p6) vector meson models contribut-ing to this channel and to KL → π◦γγ showing that local contributions generated by vectormeson exchange in the charged channel are likely to be negligible contrarily to the neutralchannel. This can be studied in the spectrum for small diphoton invariant mass, while theO(p4) unknown local contribution can be determined from the rest of the kinematical region,or from the rate.

References[1] G. Ecker, A. Pich, E. de Rafael, Phys. Lett., B189 (1987) 363.L. Cappiello, G. D’Ambrosio, Nuovo Cimento, 99A (1988) 155.[2] G. D’Ambrosio, G. Ecker, G. Isidori, H. Neufeld, “Radiative non–leptonic kaon decays” inthe Second DAΦNE Physics Handbook, ed. by L. Maiani, G. Pancheri, N. Paver, LNF (1995),p.265.[3] G. Ecker, A. Pich, E. de Rafael, Nucl. Phys., B303 (1988) 665.[4] G. D’Ambrosio, J. Portoles, Preprint INFNNA–IV–96/12, hep-ph 9606213.[5] G. D’Ambrosio, J. Portoles, Preprint INFNNA–IV–96/21.

43


Radiative Four–Meson Amplitudes in CHPT

Gino IsidoriINFN, Laboratori Nazionali di Frascati, P.O. Box 13, I–00044 Frascati, Italy

Chiral perturbation theory (CHPT) [1,2,3] naturally incorporates electromagnetic gaugeinvariance. To lowest order in the derivative expansion, O(p2) in the meson sector, amplitudesfor radiative transitions are completely determined by the corresponding non–radiative ampli-tudes. Direct emission (DE), carrying genuinely new information, appears only at O(p4). Inthe case of K → 2πγ and K → 3πγ decays, the study of these information is of great interestto understand the structure of the O(p4) nonleptonic weak lagrangian [4,5].

The fact that O(p2) radiative amplitudes are completely determined by the correspondingnon–radiative ones is a consequence of Low’s theorem [6]. In the case of radiative three–mesonprocesses, like K → 2πγ decays, where the on–shell non–radiative amplitude is constant, it isstraightforward to extend the relation between radiative and non–radiative amplitudes to higherorders in the chiral expansion [7,8,9,10]. On the other hand, in the case of radiative four–mesonprocesses, the dependence from kinematical variables of the non–radiative amplitudes makesthis extension less trivial. It has been shown in Ref. [11] how to extend Low’s theorem by meansof second derivatives of the non–radiative amplitudes to define a “generalized bremsstrahlung”(GB). This amplitude include all the contributions to the radiative process generated by localO(p4) counterterms that contribute to the non–radiative one. By this way, the remaining part ofthe radiative amplitude receives O(p4) contributions only form genuine radiative counterterms(operators with an explicit electromagnetic strength tensor) and from loop diagrams.

In principle, the GB can be calculated using the experimental information on the nonradiative process, minimizing the uncertainties related to higher order effects in CHPT. Onthe other hand, the remaining contributions must be computed using O(p4) CHPT predictions.The only loop diagrams that contribute to the DE, i.e. which are not included in the GB,are the so–called “fish–diagrams”. In Ref. [11] a compact but completely general expressionfor these loop amplitudes has been presented. Using the most general parametrization of theO(p2) four–meson vertices, the loop amplitudes of Ref. [11] can be applied to any radiativefour–meson process, both in the strong and in the weak sector (known results for K → 2πγdecays [7,8,9,10] are recovered as a particular case).

Detailed numerical analysis for K → 3πγ and η → 3πγ transitions are in progress [12].References

[1] S. Weinberg, Physica 96A (1979) 327.[2] J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142.[3] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465.[4] J. Kambor, J. Missimer and D. Wyler, Nucl. Phys. B346 (1990) 17.[5] G. Ecker, J. Kambor and D. Wyler, Nucl. Phys. B394 (1993) 101.[6] F.E. Low, Phys. Rev. 110 (1958) 974.[7] G. Ecker, H. Neufeld and A. Pich, Phys. Lett. B278 (1992) 337.[8] G. D’Ambrosio, M. Miragliuolo and F. Sannino, Z. Phys. C59 (1993) 451.[9] G. Ecker, H. Neufeld and A. Pich, Nucl. Phys. B413 (1994) 321.[10] G. D’Ambrosio and G. Isidori, Z. Phys. C65 (1995) 649.[11] G. D’Ambrosio, G. Ecker, G. Isidori and H. Neufeld, Preprint INFNNA-IV-96/11 [hep-ph/9603345], to appear in Phys. Lett. B[12] G. D’Ambrosio, G. Ecker, G. Isidori and H. Neufeld, in preparation.

44



Aspects of Renormalization in Chiral Perturbation Theory

Gerhard EckerInst. Theor. Physik, Univ. Wien, Vienna, Austria

In a short introduction to the loop expansion in CHPT, I discussed the advantages for aconsistent chiral power counting of using the lowest–order mesonic chiral Lagrangian of O(p2)to define the classical solution as the starting point for the loop expansion to any chiral order.One important implication is that the lowest–order equation of motion can be used in thegenerating functional for any chiral order. The renormalization procedure for the ππ scatteringamplitude as well as for Mπ and Fπ to O(p6) [1] was the main topic of my presentation atthe Workshop. Two–loop, one–loop and tree–level diagrams add up to the final renormalizedquantities. Various consistency checks for the calculation were discussed that are essentiallydue to the proper handling of subdivergences of O(p4). One of these conditions allows forthe calculation of the leading squares of chiral logs appearing at O(p6) in terms of one–loopdiagrams only (with a single O(p4) vertex) [2,3]. In the last part, I analysed the mesonicgenerating functional of O(p4) with one off–shell meson line to arrive at the following generalconclusions:

1. In the calculation of the meson–baryon functional of O(p3) [4,5] in heavy–baryon CHPT,the relative contributions of the local meson–baryon action and of the reducible tree–leveldiagrams with one vertex from the mesonic Lagrangian of O(p4) depend on the choice ofthe latter Lagrangian, i.e. on the choice of meson fields. The sum is of course independentof the chosen convention.

2. Expanding the mesonic low–energy constants of O(p4) in a Laurent expansion aroundd = 4, the coefficients linear in d − 4 appear in general in mesonic amplitudes of O(p6)due to two–loop diagrams. These terms can always be absorbed in the coupling constantsof O(p6) in a process independent fashion. In other words, those coefficients are notmeasurable quantities independent of the low–energy constants of O(p6).

References[1] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M.E. Sainio, Phys. Lett. B374 (1996)210; more detailed version in preparation.[2] S. Weinberg, Physica 96A (1979) 327.[3] G. Colangelo, Phys. Lett. B350 (1995) 85; ibid. B361 (1995) 234 (E).[4] G. Ecker, Phys. Lett. B336 (1994) 508.[5] G. Ecker and M. Mojzis, Phys. Lett. B365 (1996) 312.

45

Chiral Symmetry and Hypernuclei

Roxanne P. SpringerDuke University

Durham, North Carolina, USA

Hypernuclei provide another laboratory for testing predictions of heavy baryon chiral pertur-bation theory[1]. In large-A nuclei, the free Λ → Nπ mesonic decay is Pauli blocked. Instead,the hypernucleus decays through the (nonmesonic) reaction ΛN → NN. The meson exchangecontribution to this process is dominated by pion exchange, where both the weak and strongvertices required can be found experimentally. The total nonmesonic decay widths are wellreproduced using a variety of models, while the ratio of proton induced to neutron induceddecays is much more difficult to understand. Shell model calculations on 12

Λ C[2] indicate thatthe kaon meson exchange plays an important role in such ratios. We calculate the leading SU(3)breaking one-loop corrections to the weak KNN couplings relevant for this decay [3]. One ofthe motivations for this calculation is to further investigate the hyperon p-wave problem. Formany years it has been known that the chiral coefficients dictated by the s-wave hyperon decaysreproduce the p-wave data very poorly. Further, the leading logarithmic loop calculations forthis process are found to be large, yet still in severe disagreement with the data [4,5]. Thisfinding led to concerns about the validity of chiral perturbation theory for this process [6]. Thep-wave KNN couplings that we calculate arise from the same set of diagrams which correct thep-wave hyperon decays. Therefore, a comparison of this calculation with data extracted fromhypernuclear decays may lead to a better understanding of what should be expected from chiralperturbation theory in this sector. We find that the leading logarithmic corrections to KNNcouplings are well behaved. This supports the suggestion that the problem in hyperon p-wavepredictions comes from accidental cancellations of tree level diagrams rather than problemsinherent in the theory[5]. The values for the weak KNN couplings that we find are smallerthan the tree-level values. These couplings will now be used in a shell model calculation to testagreement with experimental obervables[7].

References[1] E.Jenkins and A.Manohar, ”Baryon Chiral Perturbation Theory,” presented at Hungary,August, 1991.[2] C.Bennhold, A.Parreno, A.Ramos, Few-Body Systems Suppl. (1996) 1.[3] M.J.Savage and R.P.Springer, Phys. Rev. C53 (1996) 441.[4] J.Bijnens, H.Sonoda, and M.B.Wise, Nucl. Phys. B261 (1985) 185.[5] E.Jenkins, Nucl. Phys. B375 (1992) 561.[6] C.Carone and H.Georgi, Nucl. Phys. B375 (1992) 243.[7] C.Bennhold, private communication.

46

Hyperon Electromagnetic Properties in a Soliton Model

Norberto N. Scoccola†

INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy.

The predictions obtained within the bound state soliton model[1] for the electromagnetic decaywidths of the decuplet hyperons, the electromagnetic decay widths of the Λ(1405) resonanceand the electric and magnetic static polarizabilities of the octet hyperons are discussed. Detailsof this work are given in Refs.[2].

Our results for the radiative decay widths of the decuplet hyperons are in good agreementwith those obtained using the non-relativistic quark model (NRQM), the bag model, heavybaryon chiral perturbation theory (HBChPT) and quenched lattice QCD. This overall agree-ment between different models contrasts with the situation for the Λ(1405) decay widths. There,our predictions agree rather well with the results of the cloudy bag model but are, however,much smaller than those of the NRQM. Concerning the static electric polarizabilities we obtainrather small splittings between the values corresponding to the different hyperons. Moreover,they are dominated by the seagull terms which are basically given by the non-strange contri-butions. The structure is richer in the magnetic case because of the interplay between a large(negative) seagull part with the relevant dispersive contribution. Although some of our resultsare in agreement with those of the NRQM, in general this is not the case. In addition, thecalculations performed in the framework of HBChPT lead to still different predictions. In thissituation, it is clear that the future experimental data from CEBAF and FNAL could be ofgreat help to discriminate among the different existing models of hyperons.

References[1] C.G. Callan and I. Klebanov, Nucl. Phys. B262 (1985) 365; N.N. Scoccola, H. Nadeau,M.A. Nowak and M. Rho, Phys. Lett. B201 (1988) 425; C.G. Callan, K. Hornbostel and I.Klebanov, Phys. Lett. B202 (1988) 269; U. Blom, K. Dannbom and D.O. Riska, Nucl. Phys.A493 (1989) 384.[2] C.L. Schat, C. Gobbi and N.N. Scoccola, Phys. Lett. B356 (1995) 1; C.L. Schat, N.N.Scoccola and C. Gobbi, Nucl. Phys. A585 (1995) 627; C. Gobbi, C.L. Schat and N.N. Scoccola,Nucl. Phys. A598 (1996) 318.

†On leave from Physics Dept., CNEA, Argentina and Fellow of the CONICET, Argentina.

47

Strange contents in the nucleon

– The effects of kaonic cloud –– on the neutron electric form factor –

Teruaki Watabe and Klaus GoekeInstitute for Theoretical Physics II, Ruhr-University Bochum,

D-44780 Bochum, Federal Republic Germany

As well known the nucleon is constructed by three quarks. For instance, the neutron hasone up quark, which has a charge of +2

3, and two down quarks, with a charge of −1

3. The

experiments indicate that the nucleon has an abundant structure measuring its form factor andmagnetic moment. The electric charge distribution of the nucleon comes from the bare valencequarks (qqq) and valence quarks with the mesonic excitation of the vacuum (qqq′ + q′q). Thetotal charge of neutron is zero, therefore the bare valence quark contribution to the electriccharge distribution is nearly zero and the neutron electric charge distribution is dominated bythe mesonic excitation of the vacuum. Hence one can expect that mesonic clouds play quiteimportant role on the electric properties of the neutron. We investigate them employing thechiral quark soliton model. The chiral quark soliton (χQS) model provides a well-defined andreliable framework in studying effects of mesonic clouds on the nucleon properties. The χQSmodel is derived based on the instanton picture of the QCD vacuum [1] and described by avery simple QCD effective action, in which quarks interact via Goldstone bosons. The mesonicclouds in the χQS model are quite distinguished from other hedgehog models, since they aregenerated by the Dirac-sea quark polarization via one-quark loops.

The naive evaluation of the neutron electric form factor with the hedgehog pion whichhas the Yukawa tail behavior characterized by the pion mass gives a serious underestimation,because the kaon field arises as the rotational excitation of the hedgehog pion field which hasthe same tail behavior as the pion field. We have solved this problem using the hybrid method

of treating the mesonic clouds [2]. The neutron electric form factor is quite sensitive to themesonic clouds and the result of hybrid method fairly agrees with the experiments. We haveshown also the strange electric form factor and the square radius using the hybrid method andobtained remarkably smaller results than those appearing in the previous works done in thesame model framework. We have investigated also the proton electric properties, however theyare rather insensitive to the mesonic clouds.

References[1] D.I. Diakonov and V.Yu. Petrov, Nucl.Phys. B272 (1986) 457.[2] T. Watabe, H.-C. Kim and K. Goeke, preprint; rub-tpii-17/95, e-print archive; hep-ph/9507318 (revised in May.1996), (1996).

48



χPT description of the MSM: One and two loop order

Joaquim MatiasDip. di Fisica, Universita di Padova, Via F.Marzolo 8, I-35131 Padova

χPT provide us with a general parametrization, in terms of the coefficients of the chirallagrangian, of the symmetry breaking sector of the SM at low energies. The coefficients of theoperators compatibles with the symmetries[1] (usually called chiral coefficients) can be fixedeither from the experiment, in that case we will end up with a model independent descriptionfor the dynamics of the light fields (W and Z), or from the different possible candidates asunderlying theory. In the latter case the chiral coefficients will encode the non-decouplingeffects[2] of the heavy particle/s that have been integrated out in each particular model. Dueto the good agreement of the LEP data with the MSM it seems reasonable to start by derivingthe values of the chiral coefficients aMSM

i [3] corresponding to this model. The difference oftenminute between these aMSM

i and the contribution to the chiral coefficients corresponding to theother alternative models is where the clues of what lies beyond the SM are. We consider ascenario with a Higgs large enough to allow for a mass gap in between the light and heavyparticle but sufficiently light to be able to define a perturbative series. The technique used toderive the chiral coefficients are the matching conditions between transverse connected Greenfunctions. They have been extended to include the two next-to-leading corrections[4], the one-loop 1/M2

H order and the two-loop M2H contribution, to the LEP1 relevant coefficients. It

is proposed a new formulation of the matching conditions at higher loop orders that solvesautomatically the subtleties concerning gauge invariance and gives information on the schemedependence of the chiral coefficients. As an outcome of the computation[4,5], it is shown howχPT combined with the properties of the on-shell scheme and the screening theorem of Veltmanprovide us with a powerful tool. It allows to improve the usual power counting estimation ofthe Higgs contribution[6] to the renormalized self-energies at nth-loop order by one power ofM2

H less in the Z and W renormalized self-energies and two powers less in the photon self-energy. ∆ρ, ∆r and ∆κ are also computed. All conclusions can be made extensible to anyother perturbative underlying theory (MSSM, multiHiggs models, new gauge extensions, . . . ).

References[1] A.Longhitano, Nucl. Phys. B188 (1981) 118.[2] T.Appelquist and J.Carazzone, Phys. Rev. D11 (1975) 2856.[3] M.J.Herrero and E.Ruiz Morales, Nucl. Phys. B418 (1994) 431; Nucl. Phys. B437 (1995)332; D.Espriu and J.Matias, Phys. Lett. B341 (1995) 332.[4] J.Matias, Padova Preprint DFPD 96/TH/18, hep-ph 9604390.[5] J.Matias and A.Vicini, in preparation.[6] M.B.Einhorn and J.Wudka, Phys. Rev. D39 (1989) 2758.

49


The Covariant Derivative Expansion Method(Generalized Euler Heisenberg Lagrangians)

Stephan DurrUniversity of Zurich (Switzerland)email : [email protected]

The Covariant Derivative Expansion Method (CDEM) is a powerful tool to calculate oneloop effectiveactions arising from a heavy particle which is integrated out in a given gauge field background - for aminimal coupling in the original theory as well as for a nonminimal Pauli term interaction .

In this talk emphasis is put both on the principles which make this method so efficient for itsspecific purpose and on one typical situation where it is a useful tool inside a more general effectivefield theory approach to a phenomenological problem of general interest. It is organized as follows :

In the first section, the idea of Euler Heisenberg Lagrangians is reviewed by looking at the situationin QED/QCD. It is shown that the virtual photon-photon / photon-gluon / gluon-gluon scatteringeffects due to the box diagram in lowest order result in an infinite tower of higherdimensional mixedphotonic and gluonic operators which are suppressed by increasing powers of the heavy particle massand give the best approximation of the original nonlocal vertex by a series of local ones. Up to operatorsof dimension 10 and higher the additional part in the effective action consists of one dimension 6(GGG) and several dimension 8 (FFFF,FFGG,FGGG,GGGG) operators whose coefficients are finite.For more preliminaries see e.g. ref. [1].

In the second section an exposition of the CDEM as one of the powerful tools to calculate those(finite) coefficients is given. The CDEM was developped and promoted in ref. [2] and later improvedby several autors [3]. Its main virtue is the fact that it maintains gauge invariance at every step.

The third section raises the problem of how to determine the phenomenological impact of additionalCP-violation of SM-extensions (left-right-symmetric models, multi-higgs-generalizations) generated atΛ ≥ mt by CP-violating sunset diagrams (with external gluons attached to it) at the much lowerhadronic scale µ ∼ ms. A step-by-step calculation is advocated which forces one to calculate theeffective action stemming from integrating out the bottom quark coupled to the gluonic backgroundthrough an additional CP-violating σµνG

µν term induced in the previous step.In the fourth section it is shown how the CDEM can be generalized to this situation as well. In

addition the receipe for including interactions with an electromagnetic background is given. See ref.[5] for more information.

In the fifth section an RG analysis is performed which turns the present experimental bound onthe neutron electric dipole moment into an upper bound on the bottom chromoelectric dipole momentat the scale where it arises. This is intended to be part of the answer to the question wether thepreviously mentioned high energy theories are already in conflict with present data . See ref. [4] formore information.

References

[1] V.A. Novikov etal, Physics Reports C41 (1978) Chap.6 .[2] V.A. Novikov, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Sovj. Jour. Nucl. Phys. 39 (1984)77 or Fortschr. Phys. 32 (1984) 584 .[3] L.H. Chan, Phys. Rev. Lett. 57 (1986) 1199 .M.K. Gaillard, Nucl. Phys. B268 (1986) 669 .O. Cheyette, Nucl. Phys. B297 (1988) 183 .[4] D. Chang, T. Kephart, W. Keung, T. Yuan, Phys. Rev. Lett. 68 (1992) 439 .S. Durr, D. Wyler, in preparation .[5] D. Chang, T. Kephart, W. Keung, T. Yuan, Nucl. Phys. B 384 (1992) 147 .S. Durr, D. Wyler, in preparation .

50

Kaon-nucleon interaction and Λ(1405) in dense nuclear matter

T. Waas, N. Kaiser, P.B. Siegel and W. WeiseTechnische Universitat Munchen, Garching, Germany

We examine the free meson-baryon interaction in the strangeness S = −1 sector using aneffective chiral Lagrangian [1]. Potentials are derived from this Lagrangian and used in acoupled-channel calculation of the low-energy observables. The potentials are constructed suchthat in Born approximation the s-wave scattering length is the same as that given by theeffective chiral Lagrangian, up to order q2. A comparison is made with the available low-energyhadronic data of the coupled K−p, Σπ, Λπ system, which includes the Λ(1405) resonance. Goodfits to the experimental data and estimates of previously unknown Lagrangian parameters areobtained. The Λ(1405) emerges in this approach as a quasi-bound state between an antikaonand a nucleon.

Including Pauli blocking, Fermi motion and binding of the nucleons we find that the bindingforces between the antikaon and the nucleon are reduced inside nuclear matter [2]. Thereforethe Λ(1405) dissolves inside nuclear matter at higher densities. Connected with this dynamicsof the Λ(1405) is a strong non-linear density dependence of the K−p scattering amplitude innuclear matter. The real part of the K−p scattering length changes sign already at a smallfraction of nuclear matter density, less than 0.2 ρ0. This may explain the striking behavior ofthe K−-nuclear optical potential found in the analysis of kaonic atom data.

Solving the in-medium kaon dispersion relation [3], we find a strong non-linear densitydependence of the K− effective mass and decay width in symmetric nuclear matter at densitiesaround 0.1 times normal nuclear matter density ρ0. At higher densities the K− effective massdecreases slowly but stays above 0.5mK at least up to densities below 3 ρ0. In neutron matterthe K− effective mass decreases almost linearly with increasing density but remains relativelylarge (m∗

K > 0.65mK) for ρn <∼ 3 ρ0. The K+ effective mass turns out to increase very slowlywith rising density. The different behavior of K+ and K− effective mass in matter lead toobservable consequences for K± production rates in heavy ion collisions, especially for sub-threshold kinematics. Recent data taken at GSI are consistent with a lowering of K− versusK+ in-medium masses [4].

References[1] N. Kaiser, P.B. Siegel and W. Weise, Nucl. Phys. A594 (1995) 325.[2] T. Waas, N. Kaiser and W. Weise, Phys. Lett. B365 (1996) 12.[3] T. Waas, N. Kaiser and W. Weise, Phys. Lett. B (1996), in print.[4] R. Barth, E. Grosse, P. Senger et al. (KaoS collaboration), GSI report 10-95, p. 9.

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γγ → πππ and some comments on U(1)A

Johan BijnensNORDITA, Blegdamsvej 17

DK-2100 Copenhagen Ø, Denmark

My talk concentrated on three points:

1. The proces γγ → πππ is remarkable in the sense that the the next-to-leading order ismore than order of magnitude larger than the tree level predictions. Still we expect thenext-to-leading order result to be reasonably accurate.

2. It has been generally believed that the presence of the U(1)A as broken by the anomaly,makes chiral Lagrangians with a ninth Goldstone Boson rather unpredicitive. Here weshow that the functions of φ0 + θ that occur naively can always be reduced by fieldredefinitions to just a few constants. As an example, to order p2 the presence of the η′

introduces at most 4 new constants. This observation generalizes to higher order makinga proper extension of CHPT to the U(1) sector in principle feasible without recourse to1/Nc counting.

3. Another problem involving U(1)A is the fact that in the quenched approximation theη′ has a double pole. As a result quenched QCD is not a field theory. This leads tocounterintuitive results and makes the chiral limit sick. This part was to make somepropaganda for the work of Bernard, Golterman, Sharpe and collaborators. The topics Idiscussed are in [3,4].

References[1] P. Talavera et al., hep-ph/9512296, to be published in Phys. Lett. B, and work in progress.[2] J. Bijnens and M. Knecht, work in progress.[3] C. Bernard and M. Golterman, Phys. Rev. D46 (1992) 853.[4] J. Labrenz and S. Sharpe, Nucl. Phys. B(Proc. Suppl.) 34 (1994) 335, (LATTICE 93)

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arxiv:hep-ph/9606301v1 11 jun 1996 · 2018-09-21 · arxiv:hep-ph/9606301v1 11 jun 1996 ect*-96-010...

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